Domain of a Function - Examples, Exercises and Solutions

Question Types:
Domain: Exercises with fractionsDomain: Identifying and defining elementsDomain: Multiple domainsDomain of a Function: Find the domain of the functionDomain: Find the domain of an expression with fractionsDomain of a Function: Finding the Domain of a Square Root FunctionDomain of a Function: Multiple domainsDomain of a Function: Find the domain of an expression with fractions

The domain of a function includes all those values of X 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.

Detailed explanation

Suggested Topics to Practice in Advance

  1. Ways to Represent a Function
  2. Representing a Function Verbally and with Tables
  3. Graphical Representation of a Function
  4. Algebraic Representation of a Function
  5. Notation of a Function
  6. Rate of Change of a Function
  7. Variation of a Function
  8. Rate of change represented with steps in the graph of the function
  9. Rate of change of a function represented graphically
  10. Constant Rate of Change
  11. Variable Rate of Change
  12. Rate of Change of a Function Represented by a Table of Values
  13. Functions for Seventh Grade
  14. Increasing and Decreasing Intervals (Functions)
  15. Increasing functions
  16. Decreasing function
  17. Constant Function
  18. Decreasing Interval of a function
  19. Increasing Intervals of a function

Practice Domain of a Function

Question 1

Select the field of application of the following fraction:

\( \frac{8+x}{5} \)

Question 2

Select the field of application of the following fraction:

\( \frac{6}{x} \)

Question 3

Given the following function:

\( \frac{5}{x} \)

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

Question 4

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

\( \frac{9x}{4} \)

Question 5

Given the following function:

\( \frac{5+4x}{2+x^2} \)

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

Examples with solutions for Domain of a Function

Exercise #1

Select the field of application of the following fraction:

8+x5 \frac{8+x}{5}

Video Solution

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.

Answer

All numbers

Exercise #2

Select the field of application of the following fraction:

6x \frac{6}{x}

Video Solution

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≠0 x\ne0

Answer

All numbers except 0

Exercise #3

Given the following function:

5x \frac{5}{x}

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

Video Solution

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≠0 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.

Answer

Yes, x≠0 x\ne0

Exercise #4

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

9x4 \frac{9x}{4}

Video Solution

Step-by-Step Solution

Since the function's denominator equals 4, the domain of the function is all real numbers. This means that any one of the x values could be compatible.

Answer

No, the entire domain

Exercise #5

Given the following function:

5+4x2+x2 \frac{5+4x}{2+x^2}

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

Video Solution

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.

Answer

No, the entire domain

Question 1

Given the following function:

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

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

Question 2

Select the field of application of the following fraction:

\( \frac{x}{16} \)

Question 3

\( 2x+\frac{6}{x}=18 \)

What is the domain of the above equation?

Question 4

\( 2x-3=\frac{4}{x} \)

What is the domain of the exercise?

Question 5

Look at the following function:

\( \frac{20}{10x-5} \)

What is the domain of the function?

Exercise #6

Given the following function:

65(2x−2)2 \frac{65}{(2x-2)^2}

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

Video Solution

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≠0 (2x-2)^2\ne0

2x≠2 2x\ne2

x≠1 x\ne1

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

Answer

Yes, x≠1 x\ne1

Exercise #7

Select the field of application of the following fraction:

x16 \frac{x}{16}

Video Solution

Answer

All X All~X

Exercise #8

2x+6x=18 2x+\frac{6}{x}=18

What is the domain of the above equation?

Video Solution

Answer

x≠0

Exercise #9

2x−3=4x 2x-3=\frac{4}{x}

What is the domain of the exercise?

Video Solution

Answer

x≠0

Exercise #10

Look at the following function:

2010x−5 \frac{20}{10x-5}

What is the domain of the function?

Video Solution

Answer

x≠12 x\ne\frac{1}{2}

Question 1

Look at the following function:

\( \frac{2x+2}{3x-1} \)

What is the domain of the function?

Question 2

Consider the following function:

\( \frac{3x+4}{2x-1} \)

What is the domain of the function?

Question 3

Look at the following function:

\( \frac{10x-3}{5x-3} \)

What is the domain of the function?

Question 4

Look at the following function:

\( \frac{2x+2}{9x+6} \)

What is the domain of the function?

Question 5

Given the following function:

\( \frac{12}{8x-4} \)

What is the domain of the function?

Exercise #11

Look at the following function:

2x+23x−1 \frac{2x+2}{3x-1}

What is the domain of the function?

Video Solution

Answer

x≠13 x\ne\frac{1}{3}

Exercise #12

Consider the following function:

3x+42x−1 \frac{3x+4}{2x-1}

What is the domain of the function?

Video Solution

Answer

x≠12 x\ne\frac{1}{2}

Exercise #13

Look at the following function:

10x−35x−3 \frac{10x-3}{5x-3}

What is the domain of the function?

Video Solution

Answer

x≠35 x\ne\frac{3}{5}

Exercise #14

Look at the following function:

2x+29x+6 \frac{2x+2}{9x+6}

What is the domain of the function?

Video Solution

Answer

x≠−23 x\ne-\frac{2}{3}

Exercise #15

Given the following function:

128x−4 \frac{12}{8x-4}

What is the domain of the function?

Video Solution

Answer

x≠12 x\ne\frac{1}{2}

Topics learned in later sections

  1. Indefinite integral
  2. Inputing Values into a Function