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

What are Domain and Range?

In basic terms, domain and range coorespond with different values in in contrast to each other. For instance, let's check out the grade point calculation of a school where a student earns an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade adjusts with the total score. In mathematical terms, the score 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 can be stated as an instrument that catches particular items (the domain) as input and makes particular other pieces (the range) as output. This might be a machine whereby you can obtain different items for a respective quantity of money.

Here, we review 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 instance, let's check 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 batch of all input values for the function. To put it simply, it is the group of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) can be any real number because we can plug in any value for x and obtain a corresponding output value. This input set of values is necessary to discover the range of the function f(x).

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

The Range of a Function

The range of a function is the set of all possible output values for the function. To put it simply, it is the group of all y-coordinates or dependent variables. For example, applying the same function y = 2x + 1, we might see that the range will be all real numbers greater than or equivalent tp 1. Regardless of the value we assign to x, the output y will continue to be greater than or equal to 1.

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

Domain and Range in Intervals

Domain and range can also be classified with interval notation. Interval notation expresses a group of numbers applying two numbers that classify the lower and upper boundaries. For example, the set of all real numbers in the middle of 0 and 1 could be identified applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

This tells us 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 could also be represented with graphs. For example, let's review the graph of the function y = 2x + 1. Before plotting a graph, we must find 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 could see from the graph, the function is defined for all real numbers. This tells us that the domain of the function is (-∞,∞).

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

This is due to the fact that the function produces all real numbers greater than or equal to 1.

How do you determine the Domain and Range?

The process of finding domain and range values differs 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 specified for real numbers. Therefore, the domain for an absolute value function consists of 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. Therefore, each real number can be a possible input value. As the function just delivers positive values, the output of the function contains 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 oscillates among -1 and 1. Further, the function is stated for all real numbers.

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

• Domain: R.

• Range: [-1, 1]

Just see 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 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 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 Questions 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

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