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

What are Domain and Range?

To put it simply, domain and range coorespond with multiple values in in contrast to each other. For instance, 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 adjusts with the average grade. Expressed mathematically, the score is the domain or the input, and the grade is the range or the output.

Domain and range can also be thought of as input and output values. For instance, a function might be specified as a machine that catches specific items (the domain) as input and makes certain other items (the range) as output. This could be a machine whereby you could buy different items for a respective quantity of money.

Today, 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. For example, 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, 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 batch 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 might plug in any value for x and obtain itsl output value. This input set of values is required to figure out the range of the function f(x).

But, there are particular conditions under which a function must not be defined. For example, 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 batch of all possible output values for the function. To be specific, it is the set of all y-coordinates or dependent variables. So, using 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 apply to x, the output y will always be greater than or equal to 1.

But, just as with the domain, there are certain terms under which the range may not be defined. For example, 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 classified via interval notation. Interval notation explains a group of numbers working with two numbers that represent the lower and upper boundaries. For instance, the set of all real numbers among 0 and 1 could be identified using interval notation as follows:

(0,1)

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

Equally, the domain and range of a function might be represented using interval notation. So, let's consider the function f(x) = 2x + 1. The domain of the function f(x) can 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 classified with graphs. So, 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 can watch from the graph, the function is specified for all real numbers. This shows 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 figure out the Domain and Range?

The task of finding domain and range values is different for multiple types of functions. Let's take a look at some examples:

For Absolute Value Function

An absolute value function in the structure 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. For that reason, each real number might be a possible input value. As the function just returns positive values, the output of the function consists of 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. In addition, the function is stated 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 only 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 consistently 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 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 can connect you with a one on one math teacher if you are interested in assistance comprehending domain and range or the trigonometric subjects. Our Seattle math tutors are experienced educators who aim to tutor you on your schedule and tailor their teaching strategy to fit your needs. Contact us today at (206) 339-6328 to hear more about how Grade Potential can support you with reaching your educational objectives.