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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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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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