<< Chapter < Page Chapter >> Page >

Find the domain of the function f ( x ) = 5 + 2 x .

[ 5 2 , )

Got questions? Get instant answers now!

Can there be functions in which the domain and range do not intersect at all?

Yes. For example, the function f ( x ) = 1 x has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.

Using notations to specify domain and range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation    . For example, { x | 10 x < 30 } describes the behavior of x in set-builder notation. The braces { } are read as “the set of,” and the vertical bar | is read as “such that,” so we would read { x | 10 x < 30 } as “the set of x -values such that 10 is less than or equal to x , and x is less than 30.”

[link] compares inequality notation, set-builder notation, and interval notation.

Summary of notations for inequalities, set-builder, and intervals.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, , to combine two unconnected intervals. For example, the union of the sets { 2 , 3 , 5 } and { 4 , 6 } is the set { 2 , 3 , 4 , 5 , 6 } . It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

{ x |    | x | 3 } = ( , 3 ] [ 3 , )

Set-builder notation and interval notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form { x | statement about  x } which is read as, “the set of all x such that the statement about x is true.” For example,

{ x | 4 < x 12 }

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,

( 4 , 12 ]

Given a line graph, describe the set of values using interval notation.

  1. Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
  2. At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).
  3. At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).
  4. Use the union symbol to combine all intervals into one set.

Questions & Answers

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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