Definitions that Domain and Range


The domain that a function is the complete collection of possible values the the elevation variable.

You are watching: The range is the set of all values of a function will return as

In level English, this meaning means:

The domain is the collection of all feasible x-values which will certainly make the duty "work", and also will calculation real y-values.

When finding the domain, remember:

The denominator (bottom) the a fraction cannot be zero The number under a square root authorize must be positive in this section

Example 1a

below is the graph of `y = sqrt(x+4)`:

Interactive examples

Don"t miss the applet trying out these examples here:

Domain and variety interactive applet

The domain of this duty is `x ≥ −4`, since x can not be much less than ` −4`. To watch why, shot out part numbers less than `−4` (like ` −5` or ` −10`) and some an ext than `−4` (like ` −2` or `8`) in your calculator. The just ones the "work" and also give us an answer are the ones higher than or same to ` −4`. This will make the number under the square source positive.


The enclosed (colored-in) circle on the point `(-4, 0)`. This suggests that the domain "starts" in ~ this point.

How to uncover the domain

In general, we identify the domain of each function by in search of those values of the independent variable (usually x) i m sorry we space allowed come use. (Usually we need to avoid 0 on the bottom the a fraction, or an adverse values under the square source sign).


The range of a duty is the complete collection of all feasible resulting values that the dependent variable (y, usually), after we have actually substituted the domain.

In plain English, the definition means:

The variety is the resulting y-values we acquire after substituting all the possible x-values.

How to uncover the range

The range the a function is the spread out of feasible y-values (minimum y-value to maximum y-value) Substitute various x-values into the expression for y to watch what is happening. (Ask yourself: Is y constantly positive? always negative? Or probably not equal to details values?) Make certain you look for minimum and maximum worths of y. Draw a sketch! In math, it"s very true the a picture is precious a thousand words.

Example 1b

Let"s return to the example above, `y = sqrt(x + 4)`.

We notice the curve is either on or over the horizontal axis. No matter what value of x us try, us will constantly get a zero or hopeful value of y. Us say the range in this instance is y ≥ 0.

Range: `y>=0`
The curve walk on forever vertically, beyond what is presented on the graph, for this reason the variety is every non-negative worths of `y`.

Example 2

The graph the the curve y = sin x shows the range to it is in betweeen −1 and also 1.

Range: `-1
The domain the y = sin x is "all values of x", since there room no restrictions on the worths for x. (Put any kind of number into the "sin" function in your calculator. Any type of number must work, and also will give you a last answer in between −1 and also 1.)

From the calculator experiment, and also from observing the curve, we have the right to see the range is y betweeen −1 and 1. We can write this as −1 ≤ y ≤ 1.

Where did this graph come from? We learn about sin and cos graphs later on in Graphs of sin x and cos x

Note 1: Because we room assuming that just real numbers are to be provided for the x-values, numbers the lead to department by zero or come imaginary numbers (which arise native finding the square root of a an unfavorable number) room not included. The facility Numbers chapter explains much more about imagine numbers, but we do not include such number in this chapter.

Note 2: When law square source examples, many human being ask, "Don"t we get 2 answers, one positive and one negative when we discover a square root?" A square root contends most one value, not two. View this discussion: Square source 16 - how numerous answers?

Note 3: We room talking about the domain and range of functions, which have at most one y-value for each x-value, no relations (which can have much more than one.).

Finding domain and variety without using a graph

It"s always a lot much easier to work out the domain and variety when reading it off the graph (but we must make sure we zoom in and also out of the graph to make sure we see whatever we should see). However, we don"t always have accessibility to graphing software, and also sketching a graph usually needs knowing around discontinuities and so on first anyway.

As meantioned earlier, the key things to examine for are:

There are no negative values under a square root sign There space no zero values in the denominator (bottom) that a fraction

Example 3

Find the domain and variety of the role `f(x)=sqrt(x+2)/(x^2-9),` without utilizing a graph.


In the molecule (top) that this fraction, we have a square root. Come make sure the worths under the square root room non-negative, we have the right to only pick `x`-values grater 보다 or equal to -2.

The denominator (bottom) has actually `x^2-9`, which we recognise we have the right to write together `(x+3)(x-3)`. For this reason our worths for `x` cannot include `-3` (from the very first bracket) or `3` (from the second).

We don"t must worry about the `-3` anyway, because we dcided in the an initial step the `x >= -2`.

So the domain because that this case is `x >= -2, x != 3`, i m sorry we can write as `<-2,3)uu(3,oo)`.

To occupational out the range, we take into consideration top and also bottom of the portion separately.

Numerator: If `x=-2`, the top has actually value `sqrt(2+2)=sqrt(0)=0`. Together `x` rises value indigenous `-2`, the optimal will also increase (out to infinity in both cases).

Denominator: we break this up into 4 portions:

When `x=-2`, the bottom is `(-2)^2-9=4-9=-5`. We have actually `f(-2) = 0/(-5) = 0.`

Between `x=-2` and `x=3`, `(x^2-9)` it s okay closer to `0`, so `f(x)` will go come `-oo` together it gets close to `x=3`.

For `x>3`, as soon as `x` is just bigger than `3`, the worth of the bottom is simply over `0`, so `f(x)` will be a very large positive number.

For very big `x`, the optimal is large, but the bottom will certainly be much larger, for this reason overall, the role value will certainly be very small.

So we have the right to conclude the range is `(-oo,0>uu(oo,0)`.

Have a look in ~ the graph (which we attract anyway to examine we room on the right track):

Show graph

We can see in the adhering to graph the indeed, the domain is `<-2,3)uu(3,oo)` (which consists of `-2`, however not `3`), and also the range is "all values of `f(x)` except `F(x)=0`."


In general, we identify the domain by in search of those worths of the independent variable (usually x) which we room allowed to use. (We need to avoid 0 ~ above the bottom of a fraction, or an adverse values under the square root sign).

The range is uncovered by recognize the result y-values after ~ we have substituted in the possible x-values.

Exercise 1

Find the domain and selection for each of the following.

(a) `f(x) = x^2+ 2`.


Domain: The function

f(x) = x2 + 2

is identified for all genuine values that x (because there space no constraints on the value of x).

Hence, the domain of `f(x)` is

"all real values that x".

Range: Since x2 is never ever negative, x2 + 2 is never much less than `2`

Hence, the range of `f(x)` is

"all actual numbers `f(x) ≥ 2`".

We have the right to see that x can take any value in the graph, but the resulting y = f(x) worths are greater than or equal to 2.


Domain: The function


is not characterized for t = -2, as this value would an outcome in department by zero. (There would certainly be a 0 ~ above the bottom that the fraction.)

Hence the domain the f(t) is

"all real numbers except -2"

Range: No matter how large or little t becomes, f(t) will never be same to zero.

<Why? If we try to fix the equation for 0, this is what happens:


Multiply both sides by (t + 2) and also we get

`0 = 1`

This is impossible.>

So the range of f(t) is

"all actual numbers other than zero".

We deserve to see in the graph that the role is not characterized for `t = -2` and that the function (the y-values) takes all values other than `0`.

The duty `f(x)` has a domain the "all genuine numbers, `x > 2`" as defined in the question. (There space no result square roots of an adverse numbers or divisions by zero affiliated here.)

To find the range:

when `x = 2`, `f(2) = 8` when x increases from `2`, `f(x)` becomes bigger than `8` (Try substituting in some numbers to watch why.)

Hence, the range is "all real numbers, `f(x) > 8`"

Here is the graph of the function, v an open circle at `(2, 8)` indicating that the domain walk not include `x = 2` and the range does not incorporate `f(2) = 8`.

Generally, an unfavorable values of time carry out not have any meaning. Also, we need to assume the projectile access time the ground and then stops - the does no go underground.

So we should calculate when it is going come hit the ground. This will be as soon as h = 0. So us solve:

20t − 4.9t2 = 0

Factoring gives:

(20 − 4.9t)t = 0

This is true when

`t = 0\ "s"`,


`t=20/4.9 = 4.082 text(s)`

Hence, the domain the the function h is

"all real values that t such that `0 ≤ t ≤ 4.082`"

We deserve to see from the function expression the it is a parabola v its vertex facing up. (This provides sense if girlfriend think around throwing a sphere upwards. The goes up to a specific height and then falls back down.)

What is the maximum worth of h? We use the formula because that maximum (or minimum) the a quadratic function.

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The value of t that offers the preferably is

`t = -b/(2a) = -20/(2 xx (-4.9)) = 2.041 s `

So the maximum value is

20(2.041) − 4.9(2.041)2 = 20.408 m

By observing the function of h, we view that together t increases, h an initial increases to a best of 20.408 m, then h to reduce again to zero, together expected.