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f ( x ) = 3 x + 5

one-to-one

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For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

For the following exercises, graph the functions.

For the following exercises, use [link] to approximate the values.

Graph of a parabola.

If f ( x ) = −2 , then solve for x .

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If f ( x ) = 1 , then solve for x .

x = 1.8   or  or  x = 1.8

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For the following exercises, use the function h ( t ) = 16 t 2 + 80 t to find the values in simplest form.

h ( 2 ) h ( 1 ) 2 1

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h ( a ) h ( 1 ) a 1

64 + 80 a 16 a 2 1 + a = 16 a + 64

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Domain and Range

For the following exercises, find the domain of each function, expressing answers using interval notation.

f ( x ) = x 3 x 2 4 x 12

( , 2 ) ( 2 , 6 ) ( 6 , )

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f ( x ) = x 6 x 4

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Graph this piecewise function: f ( x ) = { x + 1          x < 2 2 x 3     x 2

Graph of f(x).
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Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from x = 1  to  x = 2.

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

Graph of a parabola.

increasing ( 2 , ) ; decreasing ( , 2 )

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Graph of a function.

increasing ( 3 , 1 ) ; constant ( , 3 ) ( 1 , )

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Find the local minimum of the function graphed in [link] .

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Find the local extrema for the function graphed in [link] .

local minimum ( 2 , 3 ) ; local maximum ( 1 , 3 )

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For the graph in [link] , the domain of the function is [ 3 , 3 ] . The range is [ 10 , 10 ] . Find the absolute minimum of the function on this interval.

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Find the absolute maximum of the function graphed in [link] .

Graph of a cubic function.

( 1.8 , 10 )

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Composition of Functions

For the following exercises, find ( f g ) ( x ) and ( g f ) ( x ) for each pair of functions.

f ( x ) = 4 x , g ( x ) = 4 x

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f ( x ) = 3 x + 2 , g ( x ) = 5 6 x

( f g ) ( x ) = 17 18 x ; ( g f ) ( x ) = 7 18 x

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f ( x ) = x 2 + 2 x , g ( x ) = 5 x + 1

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f ( x ) = x + 2 ,   g ( x ) = 1 x

( f g ) ( x ) = 1 x + 2 ; ( g f ) ( x ) = 1 x + 2

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f ( x ) = x + 3 2 ,   g ( x ) = 1 x

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For the following exercises, find ( f g ) and the domain for ( f g ) ( x ) for each pair of functions.

f ( x ) = x + 1 x + 4 ,   g ( x ) = 1 x

( f g ) ( x ) = 1 + x 1 + 4 x ,   x 0 ,   x 1 4

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f ( x ) = 1 x + 3 ,   g ( x ) = 1 x 9

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f ( x ) = 1 x ,   g ( x ) = x

( f g ) ( x ) = 1 x , x > 0

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f ( x ) = 1 x 2 1 ,   g ( x ) = x + 1

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For the following exercises, express each function H as a composition of two functions f and g where H ( x ) = ( f g ) ( x ) .

H ( x ) = 2 x 1 3 x + 4

sample: g ( x ) = 2 x 1 3 x + 4 ; f ( x ) = x

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H ( x ) = 1 ( 3 x 2 4 ) 3

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Transformation of Functions

For the following exercises, sketch a graph of the given function.

f ( x ) = 4 [ | x 2 | 6 ]

Graph of f(x)
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f ( x ) = ( x + 2 ) 2 1

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For the following exercises, sketch the graph of the function g if the graph of the function f is shown in [link] .

Graph of f(x)

For the following exercises, write the equation for the standard function represented by each of the graphs below.

For the following exercises, determine whether each function below is even, odd, or neither.

For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

Absolute Value Functions

For the following exercises, write an equation for the transformation of f ( x ) = | x | .

Graph of f(x).

f ( x ) = 1 2 | x + 2 | + 1

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Graph of f(x).

f ( x ) = 3 | x 3 | + 3

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For the following exercises, graph the absolute value function.

f ( x ) = | x 3 |

Graph of f(x).
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Inverse Functions

For the following exercises, find   f 1 ( x )   for each function.

f ( x ) = x x + 2

f 1 ( x ) = 2 x x 1

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For the following exercise, find a domain on which the function   f   is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of   f   restricted to that domain.

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

  1. Find   f ( g ( x ) ) and g ( f ( x ) ) .
  2. What does the answer tell us about the relationship between f ( x ) and g ( x ) ?
  1.   f ( g ( x ) ) = x and g ( f ( x ) ) = x .
  2. This tells us that f and g are inverse functions
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For the following exercises, use a graphing utility to determine whether each function is one-to-one.

f ( x ) = 1 x

The function is one-to-one.

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f ( x ) = 3 x 2 + x

The function is not one-to-one.

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If f ( 5 ) = 2 , find f 1 ( 2 ) .

5

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

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Practice test

For the following exercises, determine whether each of the following relations is a function.

y = 2 x + 8

The relation is a function.

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{ ( 2 , 1 ) , ( 3 , 2 ) , ( 1 , 1 ) , ( 0 , 2 ) }

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For the following exercises, evaluate the function f ( x ) = 3 x 2 + 2 x at the given input.

Show that the function f ( x ) = 2 ( x 1 ) 2 + 3 is not one-to-one.

The graph is a parabola and the graph fails the horizontal line test.

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Write the domain of the function f ( x ) = 3 x in interval notation.

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Given f ( x ) = 2 x 2 5 x , find f ( a + 1 ) f ( 1 ) in simplest form.

2 a 2 a

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Graph the function f ( x ) = { x + 1    if 2 < x < 3     x     if   x 3

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Find the average rate of change of the function f ( x ) = 3 2 x 2 + x by finding f ( b ) f ( a ) b a in simplest form.

2 ( a + b ) + 1

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For the following exercises, use the functions f ( x ) = 3 2 x 2 + x  and  g ( x ) = x to find the composite functions.

Express H ( x ) = 5 x 2 3 x 3 as a composition of two functions, f and g , where ( f g ) ( x ) = H ( x ) .

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For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

For the following exercises, determine whether the functions are even, odd, or neither.

f ( x ) = 5 x 2 + 9 x 6

even

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f ( x ) = 5 x 3 + 9 x 5

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Graph the absolute value function f ( x ) = 2 | x 1 | + 3.

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For the following exercises, find the inverse of the function.

f ( x ) = 3 x 5

f 1 ( x ) = x + 5 3

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

Graph of a cubic function.

On what intervals is the function increasing?

( , 1.1 )  and  ( 1.1 , )

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On what intervals is the function decreasing?

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Approximate the local minimum of the function. Express the answer as an ordered pair.

( 1.1 , 0.9 )

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Approximate the local maximum of the function. Express the answer as an ordered pair.

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

Graph of absolute function and step function.

Find f ( 2 ) .

f ( 2 ) = 2

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Write an equation for the piecewise function.

f ( x ) = { | x | if x 2 3 if x > 2

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For the following exercises, use the values listed in [link] .

x F ( x )
0 1
1 3
2 5
3 7
4 9
5 11
6 13
7 15
8 17

Solve the equation F ( x ) = 5.

x = 2

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Is the graph increasing or decreasing on its domain?

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Is the function represented by the graph one-to-one?

yes

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Find F 1 ( 15 ) .

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Given f ( x ) = 2 x + 11 , find f 1 ( x ) .

f 1 ( x ) = x 11 2

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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
hello friend how are you
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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