Domain of a Function

Article Contents

1. What is the Domain?
2. When the Domain is All Real Numbers
3. Restriction: Division by Zero
4. Restriction: Square Roots
5. Combining Restrictions
6. Interval Notation
7. Reading the Domain from a Graph
8. Common Mistakes
9. Interactive Exercises

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1. What is the Domain? {#what-is-domain}

The domain of a function $f$ (written $D(f)$) is the set of all allowed inputs -- all values of $x$ for which $f(x)$ is defined and produces a real number.

Think of it this way

The domain answers: "Which values of $x$ can I plug in?"

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2. When the Domain is All Real Numbers {#domain-all-reals}

For many common functions, every real number is a valid input:

Rule: Polynomials (expressions with $x$, $x^2$, $x^3$, ...) always have domain $\mathbb{R}$.

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3. Restriction: Division by Zero {#division-by-zero}

Division by zero is undefined. If the function has $x$ in a denominator, we must exclude values that make the denominator zero.

Example 1: $f(x)=\frac{1}{x}$

The denominator is $x$. Set it equal to zero:

So $x=0$ is not allowed.

Example 2: $f(x)=\frac{3}{x-2}$

Example 3: $f(x)=\frac{x+1}{x^2-4}$

Factor the denominator:

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4. Restriction: Square Roots {#square-roots}

The expression under a square root must be non-negative (greater than or equal to zero).

Example 1: $f(x)=\sqrt{x}$

Example 2: $f(x)=\sqrt{x-3}$

Example 3: $f(x)=\sqrt{5-x}$

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5. Combining Restrictions {#combining-restrictions}

Some functions have both a fraction and a square root.

Example: $f(x)=\frac{1}{\sqrt{x-1}}$

Two conditions:

1. Square root requires: $x-1\ge 0⟹x\ge 1$

2. Denominator requires: $\sqrt{x-1}\ne 0⟹x-1\ne 0⟹x\ne 1$

Combining: $x>1$.

Example: $f(x)=\frac{\sqrt{x}}{x-4}$

1. Square root requires: $x\ge 0$
2. Denominator requires: $x\ne 4$

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6. Interval Notation {#interval-notation}

We use intervals to describe the domain concisely:

Union of intervals

When the domain has a "gap", we use $\cup$ (union):

This means all real numbers except $x=2$.

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7. Reading the Domain from a Graph {#domain-from-graph}

On a graph, the domain is the set of all $x$-values where the graph exists.

Look at the horizontal extent of the graph -- from the leftmost to the rightmost point.

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8. Common Mistakes {#common-mistakes}

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9. Interactive Exercises {#interactive-exercises}

Practice finding domains:

• Domain of Rational Functions

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Summary

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Related Articles

• Introduction to Functions -- what is a function
• Range of a Function -- which outputs are possible
• Functions and Graphs -- visualising functions
• Inverse Proportion -- a function with $D(f)=\mathbb{R}\setminus \{0\}$