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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

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het 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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