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For the following exercises, use function composition to verify that f ( x ) and g ( x ) are inverse functions.

f ( x ) = x 1 3 and g ( x ) = x 3 + 1

  f ( g ( x ) ) = x , g ( f ( x ) ) = x

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f ( x ) = 3 x + 5 and g ( x ) = x 5 3

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Graphical

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

f ( x ) = −5 x + 1

one-to-one

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For the following exercises, determine whether the graph represents a one-to-one function.

For the following exercises, use the graph of f shown in [link] .

Graph of a line.

Find f 1 ( 0 ) .

2

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Solve f 1 ( x ) = 0.

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For the following exercises, use the graph of the one-to-one function shown in [link] .

Graph of a square root function.

Sketch the graph of f 1 .

Graph of a square root function and its inverse.
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Find f ( 6 )  and  f 1 ( 2 ) .

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If the complete graph of f is shown, find the domain of f .

[ 2 , 10 ]

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If the complete graph of f is shown, find the range of f .

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Numeric

For the following exercises, evaluate or solve, assuming that the function f is one-to-one.

If f ( 6 ) = 7 , find f 1 ( 7 ) .

6

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If f ( 3 ) = 2 , find f 1 ( 2 ) .

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If f 1 ( 4 ) = 8 , find f ( 8 ) .

4

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If f 1 ( 2 ) = 1 , find f ( 1 ) .

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For the following exercises, use the values listed in [link] to evaluate or solve.

x f ( x )
0 8
1 0
2 7
3 4
4 2
5 6
6 5
7 3
8 9
9 1

Find f 1 ( 0 ) .

1

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Solve f 1 ( x ) = 7.

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Use the tabular representation of f in [link] to create a table for f 1 ( x ) .

x 3 6 9 13 14
f ( x ) 1 4 7 12 16
x 1 4 7 12 16
f 1 ( x ) 3 6 9 13 14
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Technology

For the following exercises, find the inverse function. Then, graph the function and its inverse.

f ( x ) = x 3 1

f 1 ( x ) = ( 1 + x ) 1 / 3

Graph of a cubic function and its inverse.
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Find the inverse function of f ( x ) = 1 x 1 . Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

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Real-world applications

To convert from x degrees Celsius to y degrees Fahrenheit, we use the formula f ( x ) = 9 5 x + 32. Find the inverse function, if it exists, and explain its meaning.

f 1 ( x ) = 5 9 ( x 32 ) . Given the Fahrenheit temperature, x , this formula allows you to calculate the Celsius temperature.

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The circumference C of a circle is a function of its radius given by C ( r ) = 2 π r . Express the radius of a circle as a function of its circumference. Call this function r ( C ) . Find r ( 36 π ) and interpret its meaning.

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A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, t , in hours given by d ( t ) = 50 t . Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function t ( d ) . Find t ( 180 ) and interpret its meaning.

t ( d ) = d 50 , t ( 180 ) = 180 50 . The time for the car to travel 180 miles is 3.6 hours.

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Chapter review exercises

Functions and Function Notation

For the following exercises, determine whether the relation is a function.

{ ( a , b ) , ( c , d ) , ( e , d ) }

function

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{ ( 5 , 2 ) , ( 6 , 1 ) , ( 6 , 2 ) , ( 4 , 8 ) }

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y 2 + 4 = x , for x the independent variable and y the dependent variable

not a function

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Is the graph in [link] a function?

Graph of a parabola.
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For the following exercises, evaluate the function at the indicated values: f ( 3 ) ; f ( 2 ) ; f ( a ) ; f ( a ) ; f ( a + h ) .

f ( x ) = 2 x 2 + 3 x

f ( 3 ) = 27 ; f ( 2 ) = 2 ; f ( a ) = 2 a 2 3 a ;
f ( a ) = 2 a 2 3 a ; f ( a + h ) = 2 a 2 + 3 a 4 a h + 3 h 2 h 2

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f ( x ) = 2 | 3 x 1 |

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For the following exercises, determine whether the functions are one-to-one.

Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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1+cos²A/cos²A=2cosec²A-1
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a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
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why is {2kπ} union {kπ}={kπ}? when k belong to integer
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Practice Key Terms 1

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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