<< Chapter < Page Chapter >> Page >

f ( x ) = 3 x + 5

one-to-one

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

x = 1.8   or  or  x = 1.8

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

h ( a ) h ( 1 ) a 1

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

Got questions? Get instant answers now!

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 , )

Got questions? Get instant answers now!

f ( x ) = x 6 x 4

Got questions? Get instant answers now!

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

Graph of f(x).
Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!
Graph of a function.

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Graph of a cubic function.

( 1.8 , 10 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

f ( x ) = x + 2 ,   g ( x ) = 1 x

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

Got questions? Get instant answers now!

f ( x ) = x + 3 2 ,   g ( x ) = 1 x

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

f ( x ) = 1 x + 3 ,   g ( x ) = 1 x 9

Got questions? Get instant answers now!

f ( x ) = 1 x ,   g ( x ) = x

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

Got questions? Get instant answers now!

f ( x ) = 1 x 2 1 ,   g ( x ) = x + 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Transformation of Functions

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

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

Graph of f(x)
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
Graph of f(x).

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

Got questions? Get instant answers now!

For the following exercises, graph the absolute value function.

f ( x ) = | x 3 |

Graph of f(x).
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

f ( x ) = 3 x 2 + x

The function is not one-to-one.

Got questions? Get instant answers now!

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

5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

2 a 2 a

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

f ( x ) = 3 x 5

f 1 ( x ) = x + 5 3

Got questions? Get instant answers now!

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 , )

Got questions? Get instant answers now!

On what intervals is the function decreasing?

Got questions? Get instant answers now!

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

( 1.1 , 0.9 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Write an equation for the piecewise function.

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Is the graph increasing or decreasing on its domain?

Got questions? Get instant answers now!

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

yes

Got questions? Get instant answers now!

Find F 1 ( 15 ) .

Got questions? Get instant answers now!

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

f 1 ( x ) = x 11 2

Got questions? Get instant answers now!

Questions & Answers

the diagram of the digestive system
Assiatu Reply
allimentary cannel
Ogenrwot
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 1

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask