<< Chapter < Page Chapter >> Page >

Finding the domain and range using toolkit functions

Find the domain and range of f ( x ) = 2 x 3 x .

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is ( , ) and the range is also ( , ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding the domain and range

Find the domain and range of f ( x ) = 2 x + 1 .

We cannot evaluate the function at −1 because division by zero is undefined. The domain is ( , −1 ) ( −1 , ) . Because the function is never zero, we exclude 0 from the range. The range is ( , 0 ) ( 0 , ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding the domain and range

Find the domain and range of f ( x ) = 2 x + 4 .

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

x + 4 0  when  x 4

The domain of f ( x ) is [ 4 , ) .

We then find the range. We know that f ( 4 ) = 0 , and the function value increases as x increases without any upper limit. We conclude that the range of f is [ 0 , ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the domain and range of f ( x ) = 2 x .

domain: ( , 2 ] ; range: ( , 0 ]

Got questions? Get instant answers now!

Graphing piecewise-defined functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function f ( x ) = | x | . With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude , or modulus , of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

f ( x ) = x if x 0

If we input a negative value, the output is the opposite of the input.

f ( x ) = x if x < 0

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function    is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income S would be 0.1 S if S $ 10 , 000 and $ 1000 + 0.2 ( S $ 10 , 000 ) if S > $ 10 , 000.

Piecewise function

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

f ( x ) = { formula 1     if  x  is in domain 1 formula 2     if  x  is in domain 2 formula 3     if  x  is in domain 3

In piecewise notation, the absolute value function is

| x | = { x     if   x 0 x   if   x < 0

Given a piecewise function, write the formula and identify the domain for each interval.

  1. Identify the intervals for which different rules apply.
  2. Determine formulas that describe how to calculate an output from an input in each interval.
  3. Use braces and if-statements to write the function.

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

Get the best Algebra and trigonometry course in your pocket!





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