How To Find Domain And Range Of A Graph

Finding The Range Of A Function Graphically

What Is The Domain Of Exponential Functions

The domain of an exponential function is all real numbers. The reason is that any real number is a valid input as an exponent.

This is because bx is always defined for b > 0 and x a real number. In fact, when x = 0, we get bx = b0 = 1, and f will always be a.

An exponential function f = abx is continuous, since it has no holes or vertical asymptotes .

Examples On Domain And Range

Recommended Reading: Can I Buy Domain From Google

Finding The Domain And Range Of A Function: Overview Method Examples

Finding the Domain and Range of a Function: Domain, in mathematics, is referred to as a whole set of imaginable values. These values are independent variables. In other words, in a domain, we have all the possible x-values that will make the function work and will produce real y-values. The range, on the other hand, is set as the whole set of possible yielding values of the depending variable, which in this case, is y .

Finding the domain requires determining the values of the independent variables that have been allowed to use. At the bottom of the fraction, 0 is usually debarred or we generally avoid negative values that are found under the square root sign. The range of a function is considered as an array of possible y-values. Continue reading to learn more about the domain and range of a function.

More Domain And Range Examples

In case you missed it earlier, you can see more examples of domain and range in the section Inverse Trigonometric Functions.

We fire a ball up in the air and find the height h, in metres, as a function of timet, in seconds, is given by

h = 20t 4.9t2

Find the domain and range for the functionh.

Answer

Generally, negative values of time do not have any meaning. Also, we need to assume the projectile hits the ground and then stops – it does not go underground.

So we need to calculate when it is going to hit the ground. This will be when h = 0. So we solve:

20t 4.9t2 = 0

Also Check: How To Build Website After Buying Domain

Finding Domain And Range Without Using A Graph

It’s always a lot easier to work out the domain and range when reading it off the graph . However, we don’t always have access to graphing software, and sketching a graph usually requires knowing about discontinuities and so on first anyway.

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

How Do You Write The Domain And Range

We write the domain and range of a function as the set of all the inputs a function can take and the outputs of the functions respectively. The domain and range are written from the smaller values to the larger values. The domain is written from left to right and the range is written from the top of the graph to the bottom.

Recommended Reading: How To Create A Domain Email In Google

How To Find The Domain And Range Of A Function

This article was co-authored by wikiHow Staff. Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow’s Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. This article has been viewed 208,266 times.Learn more…

Every function contains two types of variables: independent variables and dependent variables, whose values literally depend on the independent variables. For example, in the function y = f = 2x + y, x is independent and y is dependent . The valid values for a given independent variable x are collectively called the domain. The valid values for a given dependent variable y are collectively called the range.XResearch source

Find The Domain And Range Of Special Functions

Rational Function: A rational function is defined for only the non-zero values of the denominator.

Square Root Function: A square root function is defined for only the non-negative values of the expression under the radical symbol.

You May Like: How To Get In Domain

Domain And Range Of Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

For the constant function f\left=c, the domain consists of all real numbers there are no restrictions on the input. The only output value is the constant c, so the range is the set \left\ that contains this single element. In interval notation, this is written as \left, the interval that both begins and ends with c.

For the identity function f\left=x, there is no restriction on x. Both the domain and range are the set of all real numbers.

For the absolute value function f\left=|x|, there is no restriction on x. However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.

For the quadratic function f\left=^, the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.

For the cubic function f\left=^, the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.

For the cube root function f\left=\sqrt, the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative .

Determining The Domain And Range Modeled By A Linear Function

To determine the domain of a given situation, identify all possible x-values, or values of the independent variable. To determine the range of a given situation, identify all possible y-values, or values of the dependent variable.

Example 1A clown at a birthday party can blow up five balloons per minute. The relationship between the number of balloons inflated and the time that has passed can be expressed with the equation y = 5x, where x is the number of minutes passed and y is the number of balloons inflated. Find the domain and range of the relations.

In this example, the independent variable is the number of minutes. The possible x-values include all real numbers greater than or equal to 0, since time can be measured in fractional parts of a minute.

The dependent variable is the number of balloons inflated. The possible y-values include all real numbers greater than or equal to 0.

Therefore, the domain is , and the range is .

Recommended Reading: What’s The Difference Between A Website And A Domain

What Do The Symbols In Domain And Range Mean

The | means such that, the symbol means element of, and means all real numbers. Putting it all together, this statement can be read as the domain is the set of all x such that x is an element of all real numbers. The range of f = x2 in set notation is: R indicates range.

How to find the domain and range of linear and quadratic?

Summary of domain and range in tabular form: I hope that the previous example has given you the idea of how to work this out. This is a quadratic function, thus, the graph will be parabolic. I know that this will also have either a minimum or a maximum.

Which is the domain of a linear function?

The domain of a linear function is all real numbers, therefore, Domain: Range: A function with a radical . Example 2. Find the domain of the function f=2x 2 + 12x + 5. Solution. The function f = 2x 2 + 12x + 5 is a quadratic polynomial, therefore, the domain is

Domain And Range Of Trigonometric Functions

Look at the graph of the sine function and cosine function. Notice that the value of the functions oscillates between -1 and 1 and it is defined for all real numbers.

Thus, for each of the sine and cosine functions:

• Domain: The domain of the functions is the set R.
• Range: The range of the functions is

Don’t Miss: How To Create Email With Your Domain

How Do You Find The Domain And Range In Standard Form

To find the range of a standard quadratic function in the form f=ax2+bx+c, find the vertex of the parabola and determine if the parabola opens up or down. To find the vertex of a quadratic in this form, use the formula x=b2a.

How to calculate the domain and range of a function?

Calculate the domain and the range of the function f = -2/x. Set the denominator to zero. Therefore, domain: All real numbers except 0. The range is all real values of x except 0. Find the domain and range of the following function. Set the denominator equal to zero and solve for x.

Domain And Range Of An Absolute Value Function

The function y=|ax+b| is defined for all real numbers. So, the domain of the absolute value function is the set of all real numbers. The absolute value of a number always results in a non-negative value. Thus, the range of an absolute value function of the form y= |ax+b| is y R | y 0. The domain and range of an absolute value function are given as follows

Example: |6-x|

• Domain: The domain of the function is the set R.
• Range: We already know that the absolute value function results in a non-negative value always. i.e., |6-x| 0, for all x.

Read Also: How To Obtain A Org Domain

Domain And Range Of A Square Root Function

The function y= is defined only for x -b/a

So, the domain of the square root function is the set of all real numbers greater than or equal to -b/a. We know that the square root of something always results in a non-negative value. Thus, the range of a square root function is the set of all non-negative real numbers. The domain and range of a square root function are given as: Domain = [-b/a,), Range = [0,)

Example: y= 2-

Domain: A square root function is defined only when the value inside it is a non-negative number. So for a domain,

-3x+2 0

Range: We already know that the square root function results in a non-negative value always.

0

Multiply -1 on both sides

– 0

Adding 2 on both sides

2- 2

y 2

Domain And Range Of A Function Solved Examples

With all the knowledge of the domain and range, method of calculation for various types of function let us now practice some more examples to understand the same.

Solved Example 1: For the given function obtain the domain and range.

Solution:

The domain holds the set . Here D is not in the domain, as the function is not specified for D.

The range is the set . 1 is not inside the range, since no alphabet in the domain gets mapped to 1.

Solved Example 2: Find the range and domain for the graph\=\sqrt\). given below:

Solution: For the given graph function the domain is x4 as x cannot be smaller than 4.

Also, we notice that the curve is either on or beyond the horizontal axis irrespective of the value of x. Therefore the range is y 0.

Learn more about Trigonometric Ratios.

Solved Example 3:For the given below function obtain the domain and range.

Solution: For the function \ =x^\) we can have the domain of integers like ,for which the range is the set .

Solved Example 4:For the given relation obtain the domain and range.

Solution: A relation is termed as a set of ordered pairs with x and y coordinates.

Hence:

The domain of the relation is .

The range of the relation is .

Also Check: How Do I Transfer A Domain Name To Another Person

Finding Domains And Ranges

domainrange

For a particular function \\), determining \\) and \\) is sometimes quite complicated.

Example

Solution

Here a graph of the function helps.

Example

Solution

The graph of \\) is a rectangular hyperbola.

Example

Solution

This is a vertical translation of an exponential function.

Example

What is the domain and range of \=\sqrt\)?

Solution

So the graph of \\) is the top half of the circle with centre the origin and radius 4.

Hence, \ = \) and \ = \).

In the next example, we find the domain and range without first drawing the graph.

Example

Find the domain and range of \=\sqrt\).

Solution

We first find the domain of \\):

So \ = .

Every non-negative real number can be obtained as a value of \ = \sqrt\). Thus \ = [0,\infty)\).

We can sketch the graph of this function by noticing that

So this is ‘half’ of a rotated and translated parabola.

Exercise 3

Example

Find the domain and range of \=x^}\). Sketch the graph of \\).

Solution

Clearly \ = \mathbb\), since all real numbers have a cube root. As \, \\to\infty\) and as \, \\to -\infty\). Since \ is continuous, it follows that \ = \mathbb\).

We have \ if and only if \. The graph of \ is as follows.

The graph of \, or \, is obtained by reflecting in the line \.

Exercise 4

What is the domain and range of \=3\tan 2x\)? Sketch the graph of \.

Example

Find the domain and range of \=|x|\), where as usual

Solution

Exercise 5

Let \ denote the largest integer less than or equal to \. So, for example, \.

is \.)

Domain And Range From Graph

It is very easy to find the domain and range of a function if its graph is given/known. The set of values of x covered by the graph gives the domain and the set of values of y covered by the graph gives the range. But keep a note of the following things while writing the domain and range from a graph.

• See whether the graph passes the vertical line test. Otherwise, it is not a function and we do not usually define domain and range for such curves.
• If there is any hole on the graph, then its coordinates shouldn’t be in the domain and range.
• If there is a vertical asymptote, then the corresponding value of x shouldn’t be there in the domain.
• If there is a horizontal asymptote, then the corresponding value of x shouldn’t be there in the range.
• If the graph is broken into pieces, then we get multiple sets/intervals in the domain and range and we club all such sets/intervals by “union” symbol .
• If there is an arrow at the end of a curve, then it means that the curve is supposed to be extended infinitely in that particular direction.

Here is an example of a graph and we will find the domain and range from the graph.

In the above graph:

• All the x-values from – to are covered by the graph . Hence, the domain = .
• All the y-values greater than or equal than or equal to 0 are covered by the graph . Hence, the range = [0, ).

Important Notes on Domain and Range:

Related Topics:

You May Like: How To Transfer Domain Away From Go Daddy