Domain of a Function Practice Problems & Solutions

Understanding Domain of a Function

Complete explanation with examples

Domain of a Function

What is the Domain of a Function?

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.
In simple terms, the domain tells us what numbers we are allowed to plug into the function.

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.

How to Find the Domain:

The two most common cases where we encounter restrictions on the domain of a function are:

A variable in the denominator of a fraction: The denominator cannot be zero, as division by zero is undefined.
A variable under a square root or even root: The expression under the root cannot be negative, as square roots of negative numbers are not real numbers.

when we identify one (or more) of the cases, we need to solve it like we usually do, but instead of solving for the solution we'll solve to find the domain:

Variable in the Denominator:

Set the denominator not equal to zero $\text{denominator} \neq 0$ .
Solve the resulting equation to find the values to exclude from the domain.

Mathematical function F(X) = 1/X. Explanation of why X ≠ 0 due to division by zero being undefined. Fundamental algebra and function domain restriction concept.

Variable Under a Square Root or Even Root:

Set the expression inside the root greater than or equal to zero $\text{expression} \geq 0$ .
Solve the inequality to determine the allowed values for the domain.

Mathematical function F(X) = √X. Explanation that a square root cannot be negative, leading to the domain restriction X ≥ 0. Fundamental concept in algebra and function domains.

Although it might seem like most functions don’t have a specific domain, the reality is that every function has a domain. For many functions, the domain is all real numbers, meaning you can plug in any number. However, certain functions, like those with fractions or square roots, have restricted domains. for example, in this function: $f(x) = \frac{1}{x}$ the domain excludes certain numbers $like~x=0$ to avoid breaking mathematical rules.

Detailed explanation

Practice Domain of a Function

Test your knowledge with 28 quizzes

Identify the field of application of the following fraction:

$\frac{x+8}{3x}$

Examples with solutions for Domain of a Function

Step-by-step solutions included

Exercise #1

Select the domain of the following fraction:

$\frac{x}{16}$

Step-by-Step Solution

Let's examine the given expression:

$\frac{x}{16}$

As we know, the only restriction that applies to a division operation is division by 0, since no number can be divided into 0 parts, therefore, division by 0 is undefined.

Therefore, when we talk about a fraction, where the dividend (the number being divided) is in the numerator, and the divisor (the number we divide by) is in the denominator, the restriction applies only to the denominator, which must be different from 0,

However in the given expression:

$\frac{x}{16}$

the denominator is 16 and:

$16\neq0$

Therefore the fraction is well defined and thus the unknown, which is in the numerator, can take any value,

Meaning - the domain (definition range) of the given expression is:

all x

(This means that we can substitute any number for the unknown x and the expression will remain well defined),

Therefore the correct answer is answer B.

Answer:

$All~X$

Video Solution

Exercise #2

Select the domain of the following fraction:

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

Step-by-Step Solution

The domain depends on the denominator and we can see that there is no variable in the denominator.

Therefore, the domain is all numbers.

Answer:

All numbers

Video Solution

Exercise #3

Select the the domain of the following fraction:

$\frac{6}{x}$

Step-by-Step Solution

The domain of a fraction depends on the denominator.

Since you cannot divide by zero, the denominator of a fraction cannot equal zero.

Therefore, for the fraction $\frac{6}{x}$ , the domain is "All numbers except 0," since the denominator cannot equal zero.

In other words, the domain is:

$x\ne0$

Answer:

All numbers except 0

Video Solution

Exercise #4

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

What is the domain of the above equation?

Step-by-Step Solution

To solve this problem and find the domain for the expression $2x + \frac{6}{x}$ , we apply the following steps:

Step 1: Identify when the fraction $\frac{6}{x}$ is undefined. This occurs when the denominator $x$ equals zero.
Step 2: To find the restriction, set the denominator equal to zero: $x = 0$ .
Step 3: Solve for $x$ to find the values excluded from the domain. Here, $x \neq 0$ .

Since $\frac{6}{x}$ is undefined for $x = 0$ , the value $x = 0$ must be excluded from the domain.
Hence, the domain of the equation is all real numbers except zero.

Therefore, the solution to the problem, indicating the domain of the expression, is $x \neq 0$ .

Answer:

x≠0

Video Solution

Exercise #5

$2x-3=\frac{4}{x}$

What is the domain of the exercise?

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the fraction's denominator.
Step 2: Determine where this denominator equals zero.
Step 3: Exclude this value from the domain.

Now, let's work through each step:

Step 1: The given equation is $2x - 3 = \frac{4}{x}$ . Notice that the fraction $\frac{4}{x}$ has a denominator of $x$ .

Step 2: Set the denominator equal to zero to determine where it is undefined.

$\begin{aligned} x &= 0 \end{aligned}$

Step 3: Since the expression is undefined at $x = 0$ , we must exclude this value from the domain.

Therefore, the domain of the expression is all real numbers except 0, formally stated as $x \neq 0$ .

The correct solution to the problem is: x ≠ 0.

Answer:

x≠0

Video Solution

Frequently Asked Questions

Everything you need to know about Domain of a Function

What is the domain of a function and why is it important?

+

The domain of a function includes all values of the independent variable (x) that make the function valid and defined. It's crucial because it tells you exactly which numbers you can substitute into the function without breaking mathematical rules like division by zero.

How do you find the domain when there's a variable in the denominator?

+

Set the denominator not equal to zero and solve for the restricted values. For example, if f(x) = 1/(x-3), set x-3 ≠ 0, which gives x ≠ 3. The domain is all real numbers except 3.

What are the domain restrictions for square root functions?

+

For square root functions, the expression under the root must be greater than or equal to zero. Set the expression ≥ 0 and solve the inequality to find the allowed domain values.

Can a function have multiple domain restrictions?

+

Yes, functions can have multiple restrictions. For example, f(x) = √(x+1)/(x-2) has two restrictions: x+1 ≥ 0 (from the square root) and x ≠ 2 (from the denominator). The domain must satisfy both conditions.

What does it mean when we write x ≠ -2 for a domain?

+

This notation means x cannot equal -2, but can be any other real number. It's typically used when -2 would make a denominator zero or create an undefined expression in the function.

How do you write domain restrictions in interval notation?

+

Use parentheses for excluded values and brackets for included values. For x ≠ 3, write (-∞, 3) ∪ (3, ∞). For x ≥ -5, write [-5, ∞). Multiple restrictions require union symbols to combine intervals.

Why can't we have negative numbers under square roots?

+

In the real number system, square roots of negative numbers are undefined because no real number multiplied by itself gives a negative result. This creates a domain restriction requiring the expression under the root to be non-negative.

What's the domain of most polynomial functions?

+

Most polynomial functions have a domain of all real numbers because you can substitute any value for x without creating undefined expressions. Restrictions only occur with fractions, roots, or other special operations.

Continue Your Math Journey

Master Domain of a Function first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Indefinite integral Inputing Values into a Function

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

Exercises with fractions Finding the Domain of a Square Root Function Find the domain of an expression with fractions Identifying and defining elements Multiple domains Finding the Domain of a Square Root Function Find the domain of an expression with fractions Multiple domains

Domain of a Function Practice Problems & Solutions

Understanding Domain of a Function