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

What are Domain and Range?

To put it simply, domain and range refer to several values in in contrast to one another. 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 average grade. In mathematical terms, the total is the domain or the input, and the grade is the range or the output.

Domain and range might also be thought of as input and output values. For instance, a function can be defined as a machine that takes specific objects (the domain) as input and produces particular other objects (the range) as output. This might be a instrument whereby you can get several items for a particular quantity of money.

Here, we will teach you 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 cooresponds to the x-values and y-values. So, 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 batch of all input values for the function. To clarify, it is the group of all x-coordinates or independent variables. So, let's consider the function f(x) = 2x + 1. The domain of this function f(x) could be any real number because we cloud apply any value for x and obtain a corresponding output value. This input set of values is needed to figure out the range of the function f(x).

But, there are certain conditions under which a function must not be defined. For example, if a function is not continuous at a certain 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 be specific, it is the batch of all y-coordinates or dependent variables. For instance, using the same function y = 2x + 1, we might see that the range would be all real numbers greater than or equivalent tp 1. No matter what value we assign to x, the output y will always be greater than or equal to 1.

But, as well as with the domain, there are particular terms under which the range cannot be defined. 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 represented using interval notation. Interval notation expresses a batch of numbers using two numbers that represent the bottom and upper limits. For example, the set of all real numbers between 0 and 1 could 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 can be represented using interval notation. So, let's look at the function f(x) = 2x + 1. The domain of the function f(x) might be identified as follows:

(-∞,∞)

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

The range of this function can be represented as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be identified using graphs. For example, let's review the graph of the function y = 2x + 1. Before charting 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 plot these points on a coordinate plane, it will look like this:

As we could watch 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,∞).

That’s because the function produces 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 differs for different types of functions. Let's take a look at some examples:

For Absolute Value Function

An absolute value function in the form y=|ax+b| is specified for real numbers. For that reason, the domain for an absolute value function includes 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, each real number could 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 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]

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 form y= √(ax+b) is specified just for x ≥ -b/a. Consequently, 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 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

Let Grade Potential Help You Excel With Functions

Grade Potential would be happy to match you with a one on one math instructor if you need assistance understanding domain and range or the trigonometric topics. Our Charlotte math tutors are practiced educators who focus on partner with you on your schedule and customize their teaching strategy to suit your needs. Call us today at (704) 610-6275 to hear more about how Grade Potential can assist you with achieving your learning goals.