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

What are Domain and Range?

In simple terms, domain and range coorespond with different values in comparison to each other. For instance, let's consider the grade point calculation of a school where a student gets an A grade for an average between 91 - 100, a B grade for an average between 81 - 90, and so on. Here, the grade changes with the average grade. Expressed mathematically, the total 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 stated as a machine that catches specific objects (the domain) as input and generates certain other items (the range) as output. This can be a tool whereby you could obtain several snacks for a specified quantity of money.

In this piece, we review 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 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 batch of all x-coordinates or independent variables. So, let's review the function f(x) = 2x + 1. The domain of this function f(x) might 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 find the range of the function f(x).

However, there are specific conditions under which a function must not be stated. For example, if a function is not continuous at a specific point, then it is not defined for that point.

The Range of a Function

The range of a function is the batch of all possible output values for the function. In other words, it is the batch of all y-coordinates or dependent variables. For instance, applying the same function y = 2x + 1, we might see that the range is all real numbers greater than or equal to 1. Regardless of the value we apply to x, the output y will always be greater than or equal to 1.

Nevertheless, just like with the domain, there are particular conditions under which the range may not be defined. For example, 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 represented with interval notation. Interval notation indicates a batch of numbers applying two numbers that classify the lower and higher boundaries. For example, the set of all real numbers among 0 and 1 might be identified working with interval notation as follows:

(0,1)

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

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

(-∞,∞)

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

The range of this function might be identified as follows:

(1,∞)

Domain and Range Graphs

Domain and range might also be classified with graphs. So, let's consider the graph of the function y = 2x + 1. Before creating a graph, we must discover 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 graph 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 tells us that the domain of the function is (-∞,∞).

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

That’s because 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 is different for different types of functions. Let's watch some examples:

For Absolute Value Function

An absolute value function in the structure y=|ax+b| is defined for real numbers. Consequently, 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. For that reason, every real number could be a possible input value. As the function only returns 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 alternates among -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]

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 form y= √(ax+b) is stated only 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 will consistently 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

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Grade Potential can set you up with a private math instructor if you are interested in assistance comprehending domain and range or the trigonometric subjects. Our Bronx math tutors are experienced professionals who strive to partner with you when it’s convenient for you and tailor their teaching techniques to fit your needs. Reach out to us today at (718) 395-5954 to hear more about how Grade Potential can help you with achieving your educational objectives.