Domain and Range - Examples | Domain and Range of a Function

What are Domain and Range?

In simple terms, domain and range coorespond with several values in comparison to each other. For instance, let's take a look at grade point averages of a school where a student receives an A grade for a cumulative score of 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade changes with the average grade. In mathematical terms, the score is the domain or the input, and the grade is the range or the output.

Domain and range might also be thought of as input and output values. For instance, a function might be stated as a machine that takes particular objects (the domain) as input and produces specific other items (the range) as output. This might be a tool whereby you might get several items for a respective amount of money.

Here, we discuss the fundamentals of the domain and the range of mathematical functions.

What are the Domain and Range of a Function?

In algebra, the domain and the range refer to the x-values and y-values. For instance, let's look at the coordinates for the function f(x) = 2x: (1, 2), (2, 4), (3, 6), (4, 8).

Here the domain values are all the x coordinates, i.e., 1, 2, 3, and 4, for the range values are all the y coordinates, i.e., 2, 4, 6, and 8.

The Domain of a Function

The domain of a function is a batch of all input values for the function. In other words, it is the batch of all x-coordinates or independent variables. For example, let's review the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we might apply any value for x and get a corresponding output value. This input set of values is necessary to discover the range of the function f(x).

Nevertheless, there are particular cases under which a function may not be specified. For example, if a function is not continuous at a certain point, then it is not specified for that point.

The Range of a Function

The range of a function is the set of all possible output values for the function. In other words, it is the group of all y-coordinates or dependent variables. So, applying the same function y = 2x + 1, we could see that the range is all real numbers greater than or the same as 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.

But, as well as with the domain, there are particular conditions under which the range may not be specified. For instance, if a function is not continuous at a specific point, then it is not stated for that point.

Domain and Range in Intervals

Domain and range could also be represented using interval notation. Interval notation expresses a batch of numbers working with two numbers that identify the lower and higher bounds. For example, the set of all real numbers among 0 and 1 might be represented applying interval notation as follows:

(0,1)

This denotes that all real numbers higher than 0 and less than 1 are included in this group.

Equally, the domain and range of a function might be represented with interval notation. So, let's review the function f(x) = 2x + 1. The domain of the function f(x) could be identified as follows:

(-∞,∞)

This reveals that the function is specified for all real numbers.

The range of this function could be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be represented via graphs. For instance, let's review the graph of the function y = 2x + 1. Before charting a graph, we need to determine all the domain values for the x-axis and range values for the y-axis.

Here are the coordinates: (0, 1), (1, 3), (2, 5), (3, 7). Once we chart these points on a coordinate plane, it will look like this:

As we might look from the graph, the function is stated for all real numbers. This shows us that the domain of the function is (-∞,∞).

The range of the function is also (1,∞).

That’s because the function produces all real numbers greater than or equal to 1.

How do you figure out the Domain and Range?

The task of finding domain and range values differs for multiple types of functions. Let's take a look at some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is specified for real numbers. For that reason, the domain for an absolute value function contains all real numbers. As the absolute value of a number is non-negative, the range of an absolute value function is y ∈ R | y ≥ 0.

The domain and range for an absolute value function are following:

• Domain: R

• Range: [0, ∞)

For Exponential Functions

An exponential function is written as y = ax, where a is greater than 0 and not equal to 1. For that reason, every real number might be a possible input value. As the function just returns positive values, the output of the function consists of all positive real numbers.

The domain and range of exponential functions are following:

• Domain = R

• Range = (0, ∞)

For Trigonometric Functions

For sine and cosine functions, the value of the function shifts among -1 and 1. Also, the function is specified for all real numbers.

The domain and range for sine and cosine trigonometric functions are:

• Domain: R.

• Range: [-1, 1]

Take a look at the table below for the domain and range values for all trigonometric functions:

For Square Root Functions

A square root function in the form y= √(ax+b) is specified only for x ≥ -b/a. Therefore, the domain of the function contains all real numbers greater than or equal to b/a. A square function will always result in a non-negative value. So, the range of the function consists of all non-negative real numbers.

The domain and range of square root functions are as follows:

• Domain: [-b/a,∞)

• Range: [0,∞)

Practice Questions on Domain and Range

Discover the domain and range for the following functions:

1. y = -4x + 3

2. y = √(x+4)

3. y = |5x|

4. y= 2- √(-3x+2)

5. y = 48

Let Grade Potential Help You Excel With Functions

Grade Potential would be happy to set you up with a private math instructor if you need assistance comprehending domain and range or the trigonometric subjects. Our San Fernando math tutors are practiced professionals who aim to partner with you when it’s convenient for you and tailor their teaching strategy to fit your needs. Call us today at (818) 946-0624 to learn more about how Grade Potential can support you with reaching your educational objectives.