# 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.

In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions
the polar co-ordinate of the point (-1, -1)
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
tanh`(x-iy) =A+iB, find A and B
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
Ajith
exponential series
Naveen
what is subgroup
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
e power cos hyperbolic (x+iy)
10y
Michael