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

What are Domain and Range?

In simple terms, domain and range apply to several values in in contrast to one another. For instance, let's take a look at the grade point calculation 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 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 example, a function might be stated as an instrument that takes specific pieces (the domain) as input and generates specific other pieces (the range) as output. This might be a tool whereby you could buy several items for a respective quantity 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 instance, let's view 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. So, let's take a look at the function f(x) = 2x + 1. The domain of this function f(x) might be any real number because we cloud apply any value for x and acquire a corresponding output value. This input set of values is required to find the range of the function f(x).

However, there are specific cases under which a function must not be specified. For instance, if a function is not continuous at a particular point, then it is not defined 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. For example, applying the same function y = 2x + 1, we could see that the range would be all real numbers greater than or equal to 1. No matter what value we assign to x, the output y will continue to be greater than or equal to 1.

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

Domain and Range in Intervals

Domain and range might also be identified using interval notation. Interval notation expresses a batch of numbers using two numbers that classify the lower and higher boundaries. For instance, the set of all real numbers among 0 and 1 can be classified working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

This tells us that the function is specified for all real numbers.

The range of this function can be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range can also be represented via graphs. For instance, let's consider the graph of the function y = 2x + 1. Before creating a graph, we need to 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 plot 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 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 determine the Domain and Range?

The process 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 might be a possible input value. As the function just returns 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 shifts between -1 and 1. Also, 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 specified just for x ≥ -b/a. Therefore, the domain of the function contains 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 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

Discover 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 Learn Functions

Grade Potential would be happy to match you with a one on one math tutor if you need help understanding domain and range or the trigonometric topics. Our Sacramento math tutors are experienced professionals who strive to tutor you when it’s convenient for you and personalize their teaching strategy to fit your learning style. Contact us today at (916) 354-6383 to hear more about how Grade Potential can support you with obtaining your learning goals.