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

What are Domain and Range?

In basic terms, domain and range coorespond with multiple values in comparison to each other. For instance, let's check out the grading system 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 total score. Expressed mathematically, the result is the domain or the input, and the grade is the range or the output.

Domain and range can also be thought of as input and output values. For instance, a function can be stated as a machine that takes respective items (the domain) as input and generates specific other items (the range) as output. This could be a instrument whereby you might buy several treats for a particular amount of money.

Today, 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 indicate the x-values and y-values. For example, 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 group of all input values for the function. To clarify, it is the group 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) could be any real number because we can plug in any value for x and obtain a respective output value. This input set of values is necessary to figure out the range of the function f(x).

Nevertheless, there are specific terms under which a function cannot be defined. So, 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 group of all possible output values for the function. To be specific, it is the group of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we might see that the range will be all real numbers greater than or equivalent tp 1. No matter what value we plug in for x, the output y will continue to be greater than or equal to 1.

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

Domain and Range in Intervals

Domain and range might also be identified with interval notation. Interval notation explains a batch of numbers applying two numbers that classify the lower and upper bounds. For example, the set of all real numbers among 0 and 1 might be represented using interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function might be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be represented via graphs. For example, let's consider the graph of the function y = 2x + 1. Before charting a graph, we must discover 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 watch from the graph, the function is stated for all real numbers. This means that the domain of the function is (-∞,∞).

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

This is because the function generates all real numbers greater than or equal to 1.

How do you find the Domain and Range?

The task of finding domain and range values is different for different types of functions. Let's consider some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is stated for real numbers. Therefore, 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 in the form of y = ax, where a is greater than 0 and not equal to 1. Consequently, every real number could be a possible input value. As the function just 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 alternates between -1 and 1. In addition, the function is specified for all real numbers.

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

• Domain: R.

• Range: [-1, 1]

Just 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. Therefore, the domain of the function consists of all real numbers greater than or equal to b/a. A square function will consistently result in a non-negative value. So, the range of the function contains 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

Find 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

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