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 22 quizzes

Given the following function:

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

What is the domain of the function?

Examples with solutions for Domain of a Function

Step-by-step solutions included

Exercise #1

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

$\frac{9x}{4}$

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

Video Solution

Exercise #2

Look at the following function:

$\frac{5}{x}$

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

Step-by-Step Solution

Since the unknown variable 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.

Answer:

Yes, $x\ne0$

Video Solution

Exercise #3

Look the following function:

$\frac{1}{5x-4}$

What is the domain of the function?

Step-by-Step Solution

To determine the domain of the function $\frac{1}{5x-4}$ , we need to find the values of $x$ for which the function is undefined. This occurs when the denominator equals zero:

First, set the denominator equal to zero:
$5x - 4 = 0$

Next, solve for $x$ :
$5x = 4$
$x = \frac{4}{5}$

The function is undefined at $x = \frac{4}{5}$ . Therefore, the domain of the function includes all real numbers except $x = \frac{4}{5}$ .

In mathematical notation, the domain is:
$x \ne \frac{4}{5}$ .

This matches choice 3 among the given options.

Answer:

$x\ne\frac{4}{5}$

Video Solution

Exercise #4

Look at the following function:

$\frac{20}{10x-5}$

What is the domain of the function?

Step-by-Step Solution

To determine the domain of the function $\frac{20}{10x-5}$ , we need to ensure that the denominator is not zero.

Step 1: Identify the denominator, which is $10x - 5$ .
Step 2: Set the denominator equal to zero and solve for $x$ . This gives us the equation:

$10x - 5 = 0$

Step 3: Add 5 to both sides of the equation:

$10x = 5$

Step 4: Divide both sides by 10 to isolate $x$ :

$x = \frac{5}{10}$

Step 5: Simplify the fraction:

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

This means that the function is undefined at $x = \frac{1}{2}$ . Therefore, the domain of the function is all real numbers except $x = \frac{1}{2}$ .

Therefore, the domain of the function is $x \ne \frac{1}{2}$ .

Answer:

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

Video Solution

Exercise #5

Given the following function:

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

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

Step-by-Step Solution

To determine the domain of the function $\frac{23}{5x-2}$ , follow these steps:

Step 1: Identify where the function is undefined by setting the denominator equal to zero.
Equation: $5x - 2 = 0$
Step 2: Solve the equation for $x$ .

Let's perform the calculation:
Step 1: Set $5x - 2 = 0$ .

Step 2: Solve for $x$ by adding 2 to both sides:
$5x = 2$

Next, divide both sides by 5:
$x = \frac{2}{5}$

This shows that the function is undefined at $x = \frac{2}{5}$ , thus excluding this point from the domain of the function.

The domain of $\frac{23}{5x-2}$ consists of all real numbers except $x = \frac{2}{5}$ .

Therefore, the domain is expressed as $x \ne \frac{2}{5}$ .

Considering the multiple-choice options, the correct choice is:

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

Answer:

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

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?

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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?

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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?

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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?

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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?

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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?

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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?

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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?

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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

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Domain of a Function Practice Problems & Solutions

Understanding Domain of a Function