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

[ 5 2 , )

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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

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
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
Aasik Reply
Wrong question
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
2x + 7 =19
2x +7=19. 2x=19 - 7 2x=12 x=6
because x is 6
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
4x=3-2 4x=1 x=1+4 x=5 5x
hi can you give another equation I'd like to solve it
what is the value of x in 4x-2+3
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
then x=-1/4
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
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
f(x)= 1350. 2^(t/20); where t is in hours.
Practice Key Terms 3

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