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

What are Domain and Range?

In basic terms, domain and range apply to different values in comparison to one another. For instance, let's consider 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 total score. In math, 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 example, a function could be stated as an instrument that takes specific objects (the domain) as input and generates certain other items (the range) as output. This can be a tool whereby you can buy different treats for a respective amount of money.

Here, we discuss the fundamentals 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. 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, because 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 set of all input values for the function. In other words, 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) 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 needed to discover the range of the function f(x).

However, there are certain cases under which a function must not be defined. For example, if a function is not continuous at a particular 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 batch of all y-coordinates or dependent variables. So, applying the same function y = 2x + 1, we can see that the range would 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.

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

Domain and Range in Intervals

Domain and range could also be classified using interval notation. Interval notation indicates a batch of numbers applying two numbers that identify the lower and higher boundaries. For example, the set of all real numbers in the middle of 0 and 1 could be represented applying interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function might be classified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be represented via graphs. So, let's review the graph of the function y = 2x + 1. Before creating 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 look from the graph, the function is defined for all real numbers. This means that the domain of the function is (-∞,∞).

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

This is due to the fact that the function creates 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 various types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is specified for real numbers. Consequently, 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. For that reason, every real number might be a possible input value. As the function only delivers 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. Also, the function is defined 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 consists of 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 consists of 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

Let Grade Potential Help You Master Functions

Grade Potential would be happy to match you with a one on one math instructor if you need support comprehending domain and range or the trigonometric topics. Our Clear Water math tutors are experienced professionals who aim to work with you on your schedule and tailor their tutoring methods to suit your learning style. Contact us today at (727) 332-0362 to learn more about how Grade Potential can support you with obtaining your learning goals.