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Domains

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Domains

Study Guide

Key Definition

The domain of a function is the set of all values for which the function is defined. For most functions in algebra, the domain is the set of all real numbers, but there are exceptions.

Important Notes

Division by zero is undefined, so values that make the denominator zero are excluded from the domain.
For functions involving square roots, the expression under the root must be non-negative.
The domain can sometimes be explicitly given as a set of values.
Real numbers include all rational and irrational numbers.
Consider the context of the problem to determine the domain.

Mathematical Notation

$\neq$ means not equal to

$\geq$ means greater than or equal to

$\leq$ means less than or equal to

$\sqrt{x}$ represents the square root of x

$\frac{a}{b}$ represents the fraction a over b

Remember to use proper notation when solving problems

Why It Works

Understanding the domain of a function helps determine where the function is valid and can be applied in real-world scenarios.

Remember

Always check the denominator and the expression under the square root when determining the domain.

Quick Reference

Domain of a Fraction:Exclude values that make the denominator zero.

Domain of a Square Root:Include values that make the expression under the root non-negative.

Understanding Domains

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The domain of a function is the set of all possible input values. For the function $f(x) = \frac{x+3}{x-2}$, the domain is all real numbers except $x \neq 2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the domain of the function $f(x) = \frac{1}{x+3}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are planning a trip and need to calculate the cost based on the number of days. The function $c(d) = \frac{500}{d-1}$ gives the cost per day. What days are possible?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the function $f(x) = \sqrt{x+4}$. Determine the domain.

Challenge Quiz

Single Choice Quiz

Advanced

What is the domain of $f(x) = \frac{\sqrt{x-1}}{x+5}$?

Recap

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