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

What are Domain and Range?

In simple terms, domain and range refer to different values in in contrast to each other. For example, let's take a look at grade point averages of a school where a student gets 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 shifts with the average grade. In mathematical terms, the total is the domain or the input, and the grade is the range or the output.

Domain and range could also be thought of as input and output values. For instance, a function might be defined as an instrument that takes specific items (the domain) as input and makes certain other items (the range) as output. This can be a machine whereby you might buy multiple treats for a respective quantity of money.

Today, we discuss the basics of the domain and the range of mathematical functions.

What is the Domain and Range of a Function?

In algebra, the domain and the range refer to the x-values and y-values. So, 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, whereas 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 group of all input values for the function. In other words, it is the set of all x-coordinates or independent variables. So, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we can plug in any value for x and acquire itsl output value. This input set of values is required to figure out the range of the function f(x).

However, there are specific cases under which a function cannot be stated. So, if a function is not continuous at a certain point, then it is not defined for that point.

The Range of a Function

The range of a function is the batch of all possible output values for the function. To be specific, it is the set of all y-coordinates or dependent variables. For example, working with the same function y = 2x + 1, we can see that the range is all real numbers greater than or equivalent tp 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.

However, just as with the domain, there are particular terms under which the range cannot be stated. For instance, if a function is not continuous at a certain point, then it is not defined for that point.

Domain and Range in Intervals

Domain and range could also be identified via interval notation. Interval notation expresses a group of numbers using two numbers that classify the lower and higher limits. For example, the set of all real numbers between 0 and 1 can be represented applying interval notation as follows:

(0,1)

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

Also, the domain and range of a function can be identified with interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) could be classified as follows:

(-∞,∞)

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

The range of this function might be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be represented with graphs. For example, let's consider 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 graph these points on a coordinate plane, it will look like this:

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

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

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

How do you find the Domain and Range?

The process of finding domain and range values differs for multiple types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is stated for real numbers. Consequently, the domain for an absolute value function includes 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 in the form of y = ax, where a is greater than 0 and not equal to 1. Consequently, each real number can be a possible input value. As the function only delivers 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 varies between -1 and 1. In addition, the function is stated 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 structure y= √(ax+b) is stated just for x ≥ -b/a. For that reason, the domain of the function includes all real numbers greater than or equal to b/a. A square function always result in a non-negative value. So, the range of the function includes all non-negative real numbers.

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

• Domain: [-b/a,∞)

• Range: [0,∞)

Practice Examples on Domain and Range

Realize 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 Master Functions

Grade Potential can set you up with a private math instructor if you are interested in assistance comprehending domain and range or the trigonometric topics. Our Greensboro math tutors are practiced educators who aim to work with you on your schedule and tailor their teaching strategy to suit your needs. Contact us today at (336) 780-7748 to learn more about how Grade Potential can help you with reaching your educational objectives.