Definitions of Domain and also Range

Domain

The domain of a function is the finish collection of feasible values of the independent variable.

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

In simple English, this definition means:

The doprimary is the collection of all feasible x-values which will make the attribute "work", and will certainly output real y-values.

When finding the domain, remember:

The denominator (bottom) of a fraction cannot be zero The number under a square root sign should be positive in this section

Example 1a

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


Interactive examples

Don"t miss out on the applet exploring these examples here:

Doprimary and also Range interenergetic applet


The domain of this attribute is `x ≥ −4`, because x cannot be much less than ` −4`. To see why, attempt out some numbers less than `−4` (prefer ` −5` or ` −10`) and some even more than `−4` (like ` −2` or `8`) in your calculator. The only ones that "work" and also offer us an answer are the ones better than or equal to ` −4`. This will make the number under the square root positive.

Notes:

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

How to find the domain

In general, we identify the domain of each feature by looking for those values of the independent variable (usually x) which we are allowed to use. (Generally we have to avoid 0 on the bottom of a portion, or negative worths under the square root sign).


Range

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

In simple English, the definition means:

The variety is the resulting y-values we obtain after substituting all the feasible x-values.

How to discover the range

The range of a duty is the spcheck out of feasible y-worths (minimum y-worth to maximum y-value) Substitute various x-worths right into the expression for y to check out what is happening. (Ask yourself: Is y always positive? Always negative? Or probably not equal to particular values?) Make sure you look for minimum and maximum worths of y. Draw a sketch! In math, it"s very true that a photo is worth a thousand also words.

Example 1b

Let"s go back to the example over, `y = sqrt(x + 4)`.

We alert the curve is either on or over the horizontal axis. No issue what worth of x we attempt, we will certainly always acquire a zero or positive worth of y. We say the range in this instance is y ≥ 0.


Range: `y>=0`
The curve goes on forever before vertically, past what is displayed on the graph, so the selection is all non-negative values of `y`.

Example 2

The graph of the curve y = sin x shows the range to be betweeen −1 and also 1.


Range: `-1
The domain of y = sin x is "all values of x", given that tbelow are no limitations on the values for x. (Placed any kind of number right into the "sin" feature in your calculator. Any number should occupational, and will certainly provide you a final answer in between −1 and also 1.)

From the calculator experiment, and from observing the curve, we can check out the range is y betweeen −1 and also 1. We can create this as −1 ≤ y ≤ 1.

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

Note 1: Since we are assuming that just genuine numbers are to be supplied for the x-values, numbers that lead to department by zero or to imaginary numbers (which aclimb from finding the square root of an unfavorable number) are not included. The Complex Numbers chapter describes even more around imaginary numbers, yet we do not include such numbers in this chapter.

Keep in mind 2: When doing square root examples, many type of human being ask, "Don"t we obtain 2 answers, one positive and one negative once we uncover a square root?" A square root contends the majority of one value, not two. See this discussion: Square Root 16 - how many type of answers?

Note 3: We are talking about the domain and also variety of functions, which have actually at most one y-worth for each x-value, not relations (which deserve to have actually more than one.).

Finding doprimary and selection without making use of a graph

It"s always a lot less complicated to work out the domain and array when reading it off the graph (however we have to make certain we zoom in and also out of the graph to make sure we check out everything we must see). However, we do not constantly have actually accessibility to graphing software, and sketching a graph usually needs understanding around discontinuities and also so on initially anymethod.

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

Tright here are no negative values under a square root authorize Tbelow are no zero worths in the denominator (bottom) of a fraction

Example 3

Find the domain and selection of the attribute `f(x)=sqrt(x+2)/(x^2-9),` without using a graph.

Solution

In the numerator (top) of this fraction, we have actually a square root. To make sure the worths under the square root are non-negative, we have the right to just select `x`-values grater than or equal to -2.

The denominator (bottom) has `x^2-9`, which we recognise we have the right to create as `(x+3)(x-3)`. So our worths for `x` cannot incorporate `-3` (from the initially bracket) or `3` (from the second).

We do not have to issue around the `-3` anymeans, bereason we dcided in the first action that `x >= -2`.

So the domain for this situation is `x >= -2, x != 3`, which we can create as `<-2,3)uu(3,oo)`.

To job-related out the range, we take into consideration optimal and also bottom of the fraction separately.

Numerator: If `x=-2`, the top has worth `sqrt(2+2)=sqrt(0)=0`. As `x` increases value from `-2`, the optimal will likewise rise (out to infinity in both cases).

Denominator: We break this up into four 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)` gets closer to `0`, so `f(x)` will go to `-oo` as it gets near `x=3`.

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

For incredibly large `x`, the top is huge, but the bottom will be much larger, so overall, the function value will certainly be incredibly little.

So we deserve to conclude the variety is `(-oo,0>uu(oo,0)`.

Have a look at the graph (which we attract anymethod to check we are on the right track):

Show graph


We deserve to see in the following graph that indeed, the domajor is `<-2,3)uu(3,oo)` (which has `-2`, however not `3`), and also the variety is "all worths of `f(x)` except `F(x)=0`."


Summary

In general, we identify the domain by looking for those values of the independent variable (commonly x) which we are allowed to usage. (We have to protect against 0 on the bottom of a portion, or negative worths under the square root sign).

The range is uncovered by finding the resulting y-values after we have actually substituted in the feasible x-worths.

Exercise 1

Find the doprimary and also range for each of the adhering to.

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

Answer


Domain: The function

f(x) = x2 + 2

is defined for all actual values of x (bereason there are no restrictions on the worth of x).

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

"all real worths of x".

Range: Due to the fact that x2 is never negative, x2 + 2 is never before less than `2`

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

"all genuine numbers `f(x) ≥ 2`".

We deserve to watch that x deserve to take any type of value in the graph, yet the resulting y = f(x) values are greater than or equal to 2.


*

Domain: The function

`f(t)=1/(t+2)`

is not characterized for t = -2, as this worth would certainly cause division by zero. (There would be a 0 on the bottom of the fractivity.)

Hence the domain of f(t) is

"all genuine numbers except -2"

Range: No matter exactly how big or small t becomes, f(t) will certainly never be equal to zero.

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

`0=1/(t+2)`

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

`0 = 1`

This is difficult.>

So the range of f(t) is

"all real numbers except zero".

We have the right to see in the graph that the function is not characterized for `t = -2` and that the function (the y-values) takes all values except `0`.


The feature `f(x)` has a domain of "all actual numbers, `x > 2`" as characterized in the question. (Tright here are no resulting square roots of negative numbers or divisions by zero associated 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 see why.)

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

Here is the graph of the function, with an open up circle at `(2, 8)` indicating that the domajor does not encompass `x = 2` and also the selection does not incorporate `f(2) = 8`.


Normally, negative values of time perform not have any kind of meaning. Also, we need to assume the projectile hits the ground and also then stops - it does not go underground.

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

20t − 4.9t2 = 0

Factoring gives:

(20 − 4.9t)t = 0

This is true when

`t = 0 "s"`,

or

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

Hence, the domain of the feature h is

"all genuine values of t such that `0 ≤ t ≤ 4.082`"

We have the right to check out from the feature expression that it is a parabola with its vertex dealing with up. (This makes feeling if you think around throwing a sphere upwards. It goes approximately a details height and then falls ago down.)

What is the maximum worth of h? We use the formula for maximum (or minimum) of a quadratic attribute.

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The value of t that offers the maximum is

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

So the maximum worth is

20(2.041) − 4.9(2.041)2 = 20.408 m

By observing the feature of h, we see that as t boosts, h first boosts to a maximum of 20.408 m, then h decreases aobtain to zero, as meant.