Algebra and Trigonometry (4th Edition)

Chapter 4

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domain, range ; The domain is the set of all ... more

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In mathematics, a function from a set X to another... more

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; Let  be the input value.Multiply   by , ... more

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; Let  be the input value.Square .Add  to the ... more

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; Let  be the input value.Subtract  from ,  .... more

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; Let  be the input value.Add  to  ,  .Take the ... more

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Multiply by , then add .  ; Use the order of ... more

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Add , then divide  by . ; Use the order of ... more

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Add , then multiply by .  ; Use the order of ... more

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Square, subtract , then divide by . ; Use the ... more

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   ; more

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  ; more

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, , , ; Replace input value with and simplify... more

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,   ,   ,    ,   ; Replace input value with and ... more

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, ,   ,    ,   ; Replace input value with and ... more

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,   ,   ,    ,       ; Replace input value  with  ... more

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,   ,   ,   ; Replace input value  with  and ... more

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,    is undefined, ,   ,   ; Replace input value  ... more

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,   ,   ,   ,   ; Replace input value  with   and ... more

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,   ,   ,    ,      ; Replace input value with  ... more

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,   ,    ,   ,   ; Replace input value  with    ... more

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,   ,   is undefined ,   ,   ; Replace input ... more

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,   ,    ,   ; Since , use .Replace input value ... more

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, ,    ,   ; Since , use .Replace input value ... more

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,   ,    ,   ; Since  , use .Replace input value ... more

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,   ,    ,    ; Since , use .Replace input value  ... more

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, ; Replace input value, , with  .    ; Multiply ... more

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,   ; Replace input value  with   and simplify... more

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,   ; Replace input value  with   and simplify... more

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; Replace input value  with  and simplify.  ; ... more

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; Determine  by replacing the input value  with  ... more

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; Determine  by replacing the input value  with  ... more

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; Determine  by replacing the input value  with  ... more

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; Determine  by replacing the input value  with  ... more

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,  , ; Replace input value, , with . ; Replace ... more

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; Replace input value, , with . ; Replace input ... more

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; Replace input value, , with . ; Replace input ... more

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, ,     ; Replace input value, , with . ;   ; ... more

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, ,   ; Replace input value, , with . ; Replace ... more

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, , ; Replace input value, , with . ; Replace ... more

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, , ; Replace input value, , with .  ; Replace ... more

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, ,  ; Replace input value, , with . ; Replace ... more

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Domain: , Range: ; Excluded values in the domain ... more

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Domain: , Range:  ; Excluded values in the domain ... more

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Domain: , Range: ; The possible values for the ... more

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Domain: , Range: ; The possible values for the ... more

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Domain: ; Determine what value of  will make the ... more

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Domain: ; Determine what value of  will make the ... more

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Domain: ; Determine what values of  will make the ... more

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Domain: ; Determine what values of  will make the ... more

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Domain: ; Determine when is greater than or equal... more

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Domain: ; Determine when is greater than or equal... more

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Domain: ; The radicand for cube roots can be any ... more

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Domain: ; Determine when is greater than or equal... more

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Domain: ; Determine when is greater than or equal... more

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Domain: ; Determine when is greater than or equal... more

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Domain: ; Determine what value of makes . ; Write... more

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Domain: ; Determine the interval over which gives... more

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Domain: ; Determine when .First factor . ; Use the... more

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Domain: ; Determine when .First factor . ; Use the... more

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Domain: ; The radicand can not be zero or negative... more

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Domain: ; The radicand can not be zero or negative... more

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Domain: ; The radicand can not be zero or negative... more

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Domain: ; The radicand can not be zero or negative... more

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Domain: , Range: ; Determine the domain. ; The ... more

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Domain: , Range: {1, all real irrational numbers... more

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The first line segment is shorter than the other ... more

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This is a rough graph. The graph starts with ... more

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This a rough graph of the given table. ; The ... more

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This a rough graph of the given table. ; The ... more

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Wages, gasoline for vehicle, sales tax on ... more

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Verbal: The employee’s wages are determined by ... more

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                           for       for    ; The... more

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, , , .  ; When graphing a function, first replace... more

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; When graphing an ordered pair of a function, ... more

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; An ordered pair on the graph of a function is ... more

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is an even power function. Thus, it is a U-shaped... more

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Fill in the table by substituting each -value into... more

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Fill in the table by substituting each -value into... more

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Fill in the table by substituting each -value into... more

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Fill in the table by substituting each -value into... more

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Generate points for the graph. Write the points as... more

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Generate points for the graph. Write the points as... more

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Generate points for the graph. Write the points as... more

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Generate points for the graph. Write the points as... more

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Generate points for the graph. Write the points as... more

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Generate points for the graph. Write the points as... more

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Generate points for the graph. Write the points as... more

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Generate points for the graph. Write the points as... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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Create a table of values with 3 columns.  First ... more

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The piecewise defined function is composed of two ... more

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The piecewise defined function is composed of two ... more

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The piecewise defined function is composed of two ... more

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The piecewise defined function is composed of two ... more

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The piecewise defined function is composed of two ... more

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The piecewise defined function is composed of two ... more

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The piecewise defined function is composed of ... more

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The piecewise defined function is composed of ... more

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The piecewise defined function is composed of two ... more

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The piecewise defined function is composed of two ... more

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Find the equivalent inequalities of the given ... more

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Find the equivalent inequalities of the given ... more

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The piecewise defined function is composed of ... more

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The piecewise defined function is composed of ... more

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Use a graphing utility to graph the given ... more

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Use a graphing utility to graph the given ... more

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; The graph in the interval is a horizontal line... more

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; The graph in the interval is a horizontal line... more

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YesDomain:  Range: ; According to the Vertical ... more

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According to the Vertical Line Test, if no ... more

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According to the Vertical Line Test, if no ... more

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YesDomain: Range: ; According to the Vertical Line... more

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Solve for :Subtract from both sides.Divide both ... more

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Solve for :Subtract from both sides.Multiply both... more

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Take the square root of both sides to solve for... more

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Solve for :Subtract from both sides.Take the ... more

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Solve for :Subtract from both sides.Divide both ... more

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Solve for :Subtract from both sides.Divide both ... more

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Write the equation in the form :Subtract from ... more

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Solve for :Add to both sides.Square both sides.... more

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Subtract from both sides to solve for .  A ... more

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Solve for :Subtract from both sides.Note that the... more

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Take the cube root of both sides to solve for .  A... more

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Solve for :Take the square root of both sides.Drop... more

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; Find the slope of the line segment with end ... more

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; Find the slope of the line segment with end ... more

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; Solve for :Subtract from both sides.Take the ... more

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; Solve for :Subtract from both sides.Take the ... more

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Graph the function using a graphing utility:Use ... more

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Graph the function using a graphing utility:Use ... more

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  ; Since, the cost of the first mile is , for . ... more

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; Find the function: The fixed cost for the first... more

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A vertical line intersects the graph of the ... more

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Example 1: The cost of tutoring service for ... more

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Shrinking the graph of horizontally by a factor ... more

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a ; The function value is the number of ticks ... more

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; By definition, the domain is the set of all ... more

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; The -axis lies where the function value, ... more

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The race was between three runners A, B, and C.... more

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; Graph using a graphing utility. ; Adjust the ... more

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; Graph using a graphing utility. ; Adjust the ... more

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; Graph using a graphing utility.The graph below... more

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Listed below are some simple real-world situations... more

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; The average speed is defined by the distance ... more

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; The average rate of change measures the slope ... more

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; Evaluate at . ; Evaluate at . ; Solve for the... more

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; Both functions started with the same value at .... more

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At , the rate of increase in the value of was the... more

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; Take the difference in the water level for the ... more

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degrees/min; degrees/min;The temperature dropped ... more

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Ave. SpeedAs the intervals get smaller, the speed ... more

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15 ; Identify two ordered pairs that lie on the ... more

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15 gal/min ; From the graph, the slope of the line... more

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Up ; A positive rate of change means that as the -... more

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YesSlope: 0Rate of change: 0 ; The given function ... more

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Yes; . ; Under the commutative property of ... more

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Yes; . ; Under the commutative property of ... more

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Not linear ; First, expand the function by ... more

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Not linear ; The variable in the function is ... more

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Yes; . ; The function can be separated into two ... more

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Not linear ; The function can be separated into ... more

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Not linear ; Simplify the given by expanding the ... more

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Yes ; Simplify the given by distributing over the ... more

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; Create a table of values for using any value ... more

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; Create a table of values for using any value ... more

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; Create a table of values for using any value ... more

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; Create a table of values for using any value ... more

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; Write the linear function where is the rate of ... more

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; Write the linear function where is the rate of ... more

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; Write the linear function as Use since the ... more

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; Write the linear function as Use since the ... more

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As increases, the line defined by becomes steeper.... more

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As increases, the graph of the line defined by ... more

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for person B; for person M ; Determine the slope... more

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; Calculate for the run.The formula of the slope ... more

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symmetry about the -axis ; In an even function, ... more

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symmetry about the origin ; In an odd function, ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to , so that . Write the function ... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Rewrite so ... more

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Let be equal to the given function. Rewrite so ... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Rewrite f(x) ... more

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Let be equal to the given function. Then, let be... more

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Let be equal to the given function. Rewrite so ... more

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; Let the final function be equal to . Since the ... more

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; Let the final function be equal to . Since the ... more

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; Let the final function be equal to . Since the ... more

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; Let the final function be equal to . Since the ... more

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; Since the graph moves to the left, add to , ... more

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; Since the graph is reflected across the -axis, ... more

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; Since the graph is reflected across the ... more

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; Since the graph moves to the left, add to , ... more

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; Multiply by to denote the vertical stretching... more

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; Multiply by to denote the vertical shrinking.... more

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; Use the vertices of the functions as reference ... more

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; Use the inflection points of the functions as ... more

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; Use the vertices of the absolute value ... more

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; Write in terms of . The graph of is ... more

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; Use the starting points of the functions as ... more

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; Use the vertices of the functions as reference ... more

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The figure shows the graph of shifted 3 units up... more

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The figure shows the graph of shifted 4 units to ... more

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Determine the horizontal span of each step in the ... more

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Determine the horizontal span of each step in the ... more

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Even ; Check if the function is even or odd. ... more

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Odd ; Check if the function is even or odd. ... more

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Neither ; Check if the function is even or odd. ... more

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Even ; Check if the function is even or odd. ... more

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Odd ; Check if the function is even or odd. ... more

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Neither ; Check if the function is even or odd. ... more

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Neither ; Check if the function is even or odd. ... more

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Odd ; Check if the function is even or odd. ... more

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The graph of is the graph of , reflected across ... more

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Identify the part of the graph of for which the ... more

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Yes; shift the function 4 units to the right and ... more

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yes; reflect the graph of across the -axis, ... more

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Case 1: The functions and are both even.Let be ... more

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Case 1: The functions and are both even.Let be ... more

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To make an even function, should be an even ... more

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8 ; Add the values of and .Read the value of on ... more

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; The small circle in between and symbolizes ... more

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multiplication by 2 comes first, then adding 1 to ... more

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; The operation for can be written as . ; The ... more

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; The composite function can be written as , ... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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Domain of :  Domain of :  Domain of :  Domain of... more

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; Separate the function into two parts.Let and... more

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; Separate the function into two parts.Let and... more

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; Express the given function as , then find the ... more

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; Separate the function into two parts.Let and... more

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To graphically add and , add a segment with the ... more

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To graphically add and , add a segment with the ... more

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Set the window settings of the graphing utility to... more

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Set the window settings of the graphing utility to... more

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Set the window settings of the graphing utility to... more

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Set the window settings of the graphing utility to... more

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4 ; Using the graph of , read the value at . ; ... more

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3 ; The given expression refers to the value of ... more

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5 ; The given expression refers to the value of ... more

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; The given expression refers to the value of ... more

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; The given expression means which refers to the... more

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; The given expression means which refers to the... more

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; The given expression refers to the value of ... more

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; The given expression refers to the value of ... more

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; The given expression refers to the value of ... more

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; The given expression refers to the value of ... more

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; Write the given expression as which refers to ... more

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; Write the given expression as which refers to ... more

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; Write the given expression as which refers to ... more

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; Write the given expression as which refers to ... more

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,  ,   ,  ,   ; Find .Substitute for the ... more

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, ,   ,   , ; Find .Substitute for the variable... more

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, , ,   , ; Find .Substitute for the variable... more

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,   , , , ; Find .Substitute for the variable... more

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,   , ,   , ; Find . Substitute for the ... more

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, ,   , , ; Find . Substitute for the variable... more

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, , ,   , ; Find . Substitute for the variable... more

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, , , , ; Find . Substitute for the variable... more

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, , , , ; Find . Substitute for every ... more

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, , , , ; Find . Substitute for the variable... more

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  ; Find . Substitute for every variable in .... more

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  ; Find . Substitute for the variable in .... more

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  ; Find . Substitute   for in . ; Find . ... more

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  ; Find . Substitute   for in . ; Find . ... more

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  ; Find . Substitute   for in . ; Find . ... more

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  ; Find .  Substitute   for in .   ; Find . ... more

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;   ; involves two operations: subtracting 9 ... more

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;   ; involves two operations: taking the square... more

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;   ; says to first square a number, and then ... more

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,  ; Observe the function :First, 3 is added to... more

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; ; says to first subtract the cube of a number ... more

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;   ; says to add one to the square root of a ... more

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; ;   ; says to first square a number, add one ... more

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; ;   ; says to first take the square root of a... more

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; ;   ; says to first take the cube root of a ... more

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; ;   ; says to first add 3 to the square root ... more

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Yes,   ; Find . Substitute for in . Simplify. ; ... more

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  ; Find . Consider as an input to a function and... more

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; Substitute for in . ; more

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; Find the revenue function by multiplying the ... more

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; The surface area of a spherical solid is given ... more

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represents the amount of investment after two ... more

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If is even, is even.If is odd and is odd, is ... more

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different ; If a function is one-to-one then for ... more

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Yes, , ; Apply the horizontal line test to the ... more

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4, 3 ; The graph of the function and the graph of... more

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No ; Apply horizontal line test to the given graph... more

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Yes ; Apply horizontal line test to the given ... more

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Yes ; Apply horizontal line test to the given ... more

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No ; Apply horizontal line test to the given graph... more

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No ; Apply horizontal line test to the given graph... more

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Yes ; Apply horizontal line test to the given ... more

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Yes ; If the function is one-to-one, then for , ... more

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Yes ; If the function is one-to-one, then for , ... more

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Yes ; The square root of a positive number is a ... more

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No ; If the function is one-to-one, then for , ... more

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No ; If the function is one-to-one, then for , ... more

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Yes ; If the function is one-to-one, then for , ... more

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No ; If the function is one-to-one, then for , ... more

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Yes ; If the function is one-to-one, then for , ... more

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Yes ; If the function is one-to-one, then for , ... more

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No ; If the function is one-to-one, then for , ... more

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No ; If the function is one-to-one, then for , ... more

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Yes ; If the function is one-to-one, then for , ... more

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1 ; Since function is one-to-one, the inverse ... more

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1 ; Since function is one-to-one, the inverse ... more

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4 ; Since function is one-to-one, the inverse ... more

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; Determine from the table by identifying the ... more

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; Determine from the table. ; Replace  by in ... more

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; Determine from the table by identifying the ... more

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; Determine from the table by identifying the ... more

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; Determine from the table by identifying the ... more

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 Since and , and are inverses of each other. ; ... more

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Since and , and are inverses of each other. ; ... more

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Since and ,   and are inverses of each other. ; ... more

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Since and , and are inverses of each other. ; ... more

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Since and , and are inverses of each other. ; ... more

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Since and , and are inverses of each other. ; ... more

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Since and , and are inverses of each other. ; ... more

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Since and , and are inverses of each other. ; ... more

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Since and , and are inverses of each other. ; ... more

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Since and , and are inverses of each other. ; ... more

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  Since and , and are inverses of each other... more

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 Since and , and are inverses of each other. ; ... more

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; Write .Solve for in terms of . ; Interchange ... more

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; Write .Solve for in terms of . ; Interchange ... more

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; Write .Solve for in terms of . ; Interchange ... more

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; Write .Solve for in terms of . ; Interchange ... more

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; Write .Solve for in terms of . ; Interchange ... more

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; Write , then solve for in terms of .Cross ... more

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; Write , then solve for in terms of .Cross ... more

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; Write , then solve for in terms of .Cross ... more

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; Write , then solve for in terms of .Cross ... more

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; Write , then solve for in terms of .Cross ... more

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; Write , then solve for in terms of .Cross ... more

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; Write , then solve for in terms of .Cross ... more

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; Write .Solve for in terms of .The domain is ... more

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; Write .Complete the square and solve for in ... more

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; Write .Solve for in terms of .The domain is ... more

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; Write .Solve for in terms of .The domain is ... more

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is the maximum value of near . ; If is a local ... more

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no ; A function has an inverse if it is one-to-one... more

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