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f ( x ) = 3 x + 5

one-to-one

f ( x ) = | x 3 |

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

Graph of a cubic function.

function

Graph of a relation.
Graph of a relation.

function

For the following exercises, graph the functions.

f ( x ) = | x + 1 |

f ( x ) = x 2 2

Graph of f(x).

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

Graph of a parabola.

f ( 2 )

f ( −2 )

2

If f ( x ) = −2 , then solve for x .

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

x = 1.8   or  or  x = 1.8

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

h ( 2 ) h ( 1 ) 2 1

h ( a ) h ( 1 ) a 1

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

Domain and Range

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

f ( x ) = 2 3 x + 2

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

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

f ( x ) = x 6 x 4

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

Graph of f(x).

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.

f ( x ) = 4 x 3

f ( x ) = 10 x 2 + x

31

f ( x ) = 2 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 )

Graph of a cubic function.
Graph of a function.

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

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

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

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

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.

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

Graph of a cubic function.

( 1.8 , 10 )

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

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

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

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

f ( x ) = x + 2 ,   g ( x ) = 1 x

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

f ( x ) = x + 3 2 ,   g ( x ) = 1 x

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

f ( x ) = 1 x + 3 ,   g ( x ) = 1 x 9

f ( x ) = 1 x ,   g ( x ) = x

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

f ( x ) = 1 x 2 1 ,   g ( x ) = x + 1

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

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

Transformation of Functions

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

f ( x ) = ( x 3 ) 2

Graph of f(x)

f ( x ) = ( x + 4 ) 3

f ( x ) = x + 5

Graph of f(x)

f ( x ) = x 3

f ( x ) = x 3

Graph of f(x)

f ( x ) = 5 x 4

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

Graph of f(x)

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

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)

g ( x ) = f ( x 1 )

Graph of a half circle.

g ( x ) = 3 f ( x )

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

Graph of an absolute function.

f ( x ) = | x 3 |

Graph of a half circle.

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

f ( x ) = 3 x 4

even

g ( x ) = x

h ( x ) = 1 x + 3 x

odd

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

Graph of a parabola.
Graph of a parabola.

even

Graph of a cubic function.

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

Graph of f(x).
Graph of f(x).

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

For the following exercises, graph the absolute value function.

Practice Key Terms 1

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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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