# 3.2 Domain and range  (Page 3/11)

 Page 3 / 11

Find the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{5+2x}.$

$\left[-\frac{5}{2},\infty \right)$

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

Yes. For example, the function $\text{\hspace{0.17em}}f\left(x\right)=-\frac{1}{\sqrt{x}}\text{\hspace{0.17em}}$ 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, $\text{\hspace{0.17em}}\left\{x|10\le x<30\right\}\text{\hspace{0.17em}}$ describes the behavior of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in set-builder notation. The braces $\text{\hspace{0.17em}}\left\{\right\}\text{\hspace{0.17em}}$ are read as “the set of,” and the vertical bar | is read as “such that,” so we would read $\text{\hspace{0.17em}}\left\{x|10\le x<30\right\}\text{\hspace{0.17em}}$ as “the set of x -values such that 10 is less than or equal to $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is less than 30.”

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

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, $\text{\hspace{0.17em}}\cup ,$ to combine two unconnected intervals. For example, the union of the sets $\left\{2,3,5\right\}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left\{4,6\right\}\text{\hspace{0.17em}}$ is the set $\text{\hspace{0.17em}}\left\{2,3,4,5,6\right\}.\text{\hspace{0.17em}}$ 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

## 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 which is read as, “the set of all $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ such that the statement about $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is true.” For example,

$\left\{x|4

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,

$\left(4,12\right]$

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 $\text{\hspace{0.17em}}\cup \text{\hspace{0.17em}}$ to combine all intervals into one set.

#### Questions & Answers

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
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find the sum of 28th term of the AP 3+10+17+---------
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Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
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write down the polynomial function with root 1/3,2,-3 with solution
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if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
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write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
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In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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the polar co-ordinate of the point (-1, -1)
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