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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

write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
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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