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Section exercises

Verbal

Why does the domain differ for different functions?

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

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How do we determine the domain of a function defined by an equation?

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Explain why the domain of f ( x ) = x 3 is different from the domain of f ( x ) = x .

There is no restriction on x for f ( x ) = x 3 because you can take the cube root of any real number. So the domain is all real numbers, ( , ) . When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x -values are restricted for f ( x ) = x to nonnegative numbers and the domain is [ 0 , ) .

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When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

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How do you graph a piecewise function?

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x -axis and y -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate or   . Combine the graphs to find the graph of the piecewise function.

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Algebraic

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

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

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f ( x ) = 5 2 x 2

( , )

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f ( x ) = 3 6 2 x

( , 3 ]

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f ( x ) = x 2 + 4

( , )

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f ( x ) = x 1 3

( , )

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

( , 1 2 ) ( 1 2 , )

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f ( x ) = x 3 x 2 + 9 x 22

( , 11 ) ( 11 , 2 ) ( 2 , )

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f ( x ) = 1 x 2 x 6

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f ( x ) = 2 x 3 250 x 2 2 x 15

( , 3 ) ( 3 , 5 ) ( 5 , )

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2 x + 1 5 x

( , 5 )

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f ( x ) = x 4 x 6

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f ( x ) = x 6 x 4

[ 6 , )

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f ( x ) = x 2 9 x x 2 81

( , 9 ) ( 9 , 9 ) ( 9 , )

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Find the domain of the function f ( x ) = 2 x 3 50 x by:

  1. using algebra.
  2. graphing the function in the radicand and determining intervals on the x -axis for which the radicand is nonnegative.
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Graphical

For the following exercises, write the domain and range of each function using interval notation.

Graph of a function from (2, 8].

domain: ( 2 , 8 ] , range [ 6 , 8 )

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Graph of a function from [-4, 4].

domain: [ 4 ,  4], range: [ 0 ,  2]

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Graph of a function from [-5, 3).

domain: [ 5 ,   3 ) , range: [ 0 , 2 ]

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Graph of a function from (-infinity, 2].

domain: ( , 1 ] , range: [ 0 , )

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Graph of a function from [-6, -1/6]U[1/6, 6]/.

domain: [ 6 , 1 6 ] [ 1 6 , 6 ] ; range: [ 6 , 1 6 ] [ 1 6 , 6 ]

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Graph of a function from [-3, infinity).

domain: [ 3 ,   ) ; range: [ 0 , )

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For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

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

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f ( x ) = { 2 x 1 if x < 1 1 + x if x 1

domain: ( , )

Graph of f(x).
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f ( x ) = { x + 1 if x < 0 x 1 if x > 0

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f ( x ) = { 3 if x < 0 x if x 0

domain: ( , )

Graph of f(x).
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f ( x ) = { x 2       if  x < 0 1 x   if  x > 0

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f ( x ) = { x 2 x + 2 if x < 0 if x 0

domain: ( , )

Graph of f(x).
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f ( x ) = { x + 1 if x < 1 x 3 if x 1

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f ( x ) = { | x | 1 if x < 2 if x 2

domain: ( , )

Graph of f(x).
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Numeric

For the following exercises, given each function f , evaluate f ( −3 ) , f ( −2 ) , f ( −1 ) , and f ( 0 ) .

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

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f ( x ) = { 1 if  x 3 0 if  x > 3

f ( 3 ) = 1 ; f ( 2 ) = 0 ; f ( 1 ) = 0 ; f ( 0 ) = 0

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f ( x ) = { 2 x 2 + 3 if  x 1 5 x 7 if  x > 1

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For the following exercises, given each function f , evaluate f ( −1 ) , f ( 0 ) , f ( 2 ) , and f ( 4 ) .

f ( x ) = { 7 x + 3 if x < 0 7 x + 6 if x 0

f ( 1 ) = 4 ; f ( 0 ) = 6 ; f ( 2 ) = 20 ; f ( 4 ) = 34

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f ( x ) = { x 2 2 if x < 2 4 + | x 5 | if x 2

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f ( x ) = { 5 x if x < 0 3 if 0 x 3 x 2 if x > 3

f ( 1 ) = 5 ; f ( 0 ) = 3 ; f ( 2 ) = 3 ; f ( 4 ) = 16

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For the following exercises, write the domain for the piecewise function in interval notation.

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

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f ( x ) = { x 2 2  if x < 1 x 2 + 2 if x > 1

domain: ( , 1 ) ( 1 , )

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f ( x ) = { 2 x 3 3 x 2 if x < 0 if x 2

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Technology

Graph y = 1 x 2 on the viewing window [ −0.5 , −0.1 ] and [ 0.1 , 0.5 ] . Determine the corresponding range for the viewing window. Show the graphs.

Graph of the equation from [-0.5, -0.1].

window: [ 0.5 , 0.1 ] ; range: [ 4 ,   100 ]

Graph of the equation from [0.1, 0.5].

window: [ 0.1 ,   0.5 ] ; range: [ 4 ,   100 ]

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Graph y = 1 x on the viewing window [ −0.5 , −0.1 ] and [ 0.1 ,   0.5 ] . Determine the corresponding range for the viewing window. Show the graphs.

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Extension

Suppose the range of a function f is [ −5 ,   8 ] . What is the range of | f ( x ) | ?

[ 0 ,   8 ]

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Create a function in which the range is all nonnegative real numbers.

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Create a function in which the domain is x > 2.

Many answers. One function is f ( x ) = 1 x 2 .

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Real-world applications

The height h of a projectile is a function of the time t it is in the air. The height in feet for t seconds is given by the function h ( t ) = −16 t 2 + 96 t . What is the domain of the function? What does the domain mean in the context of the problem?

The domain is [ 0 ,   6 ] ; it takes 6 seconds for the projectile to leave the ground and return to the ground

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The cost in dollars of making x items is given by the function C ( x ) = 10 x + 500.

  1. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.
  2. What is the cost of making 25 items?
  3. Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, C ( x ) ?
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Questions & Answers

1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
Practice Key Terms 3

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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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