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 one another. For example, let's consider the grading system of a school where a student gets an A grade for an average between 91 - 100, a B grade for a cumulative score of 81 - 90, and so on. Here, the grade changes with the total score. In math, the result 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 could be stated as a tool that catches specific pieces (the domain) as input and generates particular other items (the range) as output. This might be a tool whereby you might buy multiple items for a specified quantity of money.

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

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

The Range of a Function

The range of a function is the group of all possible output values for the function. In other words, it is the set of all y-coordinates or dependent variables. For instance, using the same function y = 2x + 1, we can see that the range will be all real numbers greater than or the same as 1. No matter what value we plug in for x, the output y will always be greater than or equal to 1.

But, just like with the domain, there are particular terms under which the range must not be specified. For instance, if a function is not continuous at a certain point, then it is not specified for that point.

Domain and Range in Intervals

Domain and range can also be classified using interval notation. Interval notation indicates a group of numbers using two numbers that identify the bottom and upper boundaries. For instance, the set of all real numbers among 0 and 1 can be identified using interval notation as follows:

(0,1)

This means that all real numbers more than 0 and lower than 1 are included in this set.

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

(-∞,∞)

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

The range of this function can be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range could also be classified using graphs. So, let's consider the graph of the function y = 2x + 1. Before creating a graph, we must 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 can see 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,∞).

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

How do you figure out 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 as y = ax, where a is greater than 0 and not equal to 1. Therefore, every real number can be a possible input value. As the function just produces positive values, the output of the function includes 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. 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 structure y= √(ax+b) is defined just for x ≥ -b/a. For that reason, 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 contains 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

Let Grade Potential Help You Master Functions

Grade Potential would be happy to match you with a private math instructor if you need help comprehending domain and range or the trigonometric topics. Our Los Angeles math tutors are practiced educators who focus on partner with you when it’s convenient for you and personalize their instruction techniques to match your learning style. Call us today at (310) 361-5334 to hear more about how Grade Potential can assist you with achieving your learning goals.