The domain of a function includes all those values of $X$ (independent variable) that, when substituted into the function, keep the function valid and defined.

The domain of a function is an integral part of function analysis. Moreover, a definition set is required to create a graphical representation of the function.

Examples with solutions for Domain of a Function

Exercise #1

Select the field of application of the following fraction:

$\frac{8+x}{5}$

Step-by-Step Solution

Since the domain depends on the denominator, we note that there is no variable in the denominator.

Therefore, the domain is all numbers.

All numbers

Exercise #2

Select the field of application of the following fraction:

$\frac{6}{x}$

Step-by-Step Solution

Since the domain of definition depends on the denominator, and X appears in the denominator

All numbers will be suitable except for 0.

In other words, the domain of definition:

$x\ne0$

All numbers except 0

Exercise #3

Given the following function:

$\frac{5}{x}$

Does the function have a domain? If so, what is it?

Step-by-Step Solution

Since the unknown is in the denominator, we should remember that the denominator cannot be equal to 0.

In other words, $x\ne0$

The domain of the function is all those values that, when substituted into the function, will make the function legal and defined.

The domain in this case will be all real numbers that are not equal to 0.

Yes, $x\ne0$

Exercise #4

Given the following function:

$\frac{9x}{4}$

Does the function have a domain? If so, what is it?

Step-by-Step Solution

Since the function's denominator equals 4, the domain of the function is all real numbers, meaning all X.

No, the entire domain

Exercise #5

Given the following function:

$\frac{65}{(2x-2)^2}$

Does the function have a domain? If so, what is it?

Step-by-Step Solution

The denominator of the function cannot be equal to 0.

Therefore, we will set the denominator equal to 0 and solve for the domain:

$(2x-2)^2\ne0$

$2x\ne2$

$x\ne1$

In other words, the domain of the function is all numbers except 1.

Yes, $x\ne1$

Exercise #6

Given the following function:

$\frac{5+4x}{2+x^2}$

Does the function have a domain? If so, what is it?

Step-by-Step Solution

Since the denominator is positive for all X, the domain of the function is the entire domain.

That is, all X, therefore there is no domain restriction.

No, the entire domain

Exercise #7

Given the following function:

$\frac{5-x}{2-x}$

Does the function have a domain? If so, what is it?

Video Solution

Yes, $x\ne2$

Exercise #8

Given the following function:

$\frac{49+2x}{x+4}$

Does the function have a domain? If so, what is it?

Video Solution

Yes, $x\ne-4$

Exercise #9

Given the following function:

$\frac{23}{5x-2}$

Does the function have a domain? If so, what is it?

Video Solution

Yes, $x\ne\frac{2}{5}$

Exercise #10

Look at the following function:

$\frac{2x+20}{\sqrt{2x-10}}$

What is the domain of the function?

x > 5

Exercise #11

Consider the following function:

$\frac{3x+4}{2x-1}$

What is the domain of the function?

Video Solution

$x\ne\frac{1}{2}$

Exercise #12

Look at the following function:

$\frac{2x+2}{3x-1}$

What is the domain of the function?

Video Solution

$x\ne\frac{1}{3}$

Exercise #13

Given the following function:

$\frac{12}{8x-4}$

What is the domain of the function?

Video Solution

$x\ne\frac{1}{2}$

Exercise #14

Look at the following function:

$\frac{5x+2}{4x-3}$

What is the domain of the function?

Video Solution

$x\ne\frac{3}{4}$

Exercise #15

Look at the following function:

$\frac{10x-3}{5x-3}$

What is the domain of the function?

Video Solution

$x\ne\frac{3}{5}$