Functions, domain and range

One input, one output

A vending machine gives one item for each code. A calendar gives one day of the week for each date. A rule that gives one and only one output for every input is a function.

Here is the exact definition. A function $f$ from a set $A$ to a set $B$ is a relation in which every element of $A$ has exactly one image in $B$. We write $f : A \to B$, and $y = f(x)$ for the image of $x$.

• $A$ is the domain. $B$ is the codomain.
• The range is the set of all the images, $\{f(x) : x \in A\}$. It is a subset of the codomain.

Which relations are functions?

Look at the arrow diagrams. In a function, every element of the domain has exactly one arrow leaving it. Two things break this rule:

• an element with two arrows, such as $1 \to 2$ and $1 \to 4$: then the image is not one number;
• an element with no arrow: then the domain is not all of $A$.

The second diagram is a function even though two elements go to the same image. Many inputs may share an output. One input may not have two outputs.

The vertical line test

On a graph, the input is $x$ and the output is $y$. A vertical line is the line of one fixed input. If the line meets the graph in two points, that input has two outputs, so the graph is not the graph of a function.

A graph is the graph of a function if and only if no vertical line meets it more than once. The circle $x^2 + y^2 = 4$ fails the test, because for $x = 1$ we get two values of $y$, $\pm\sqrt{3}$.

Real functions and their domain

A function with a domain and a range that are sets of real numbers is a real function. Often a function is given only by a formula, and we must find the largest set of $x$ for which the formula works. This is the domain. Two things can go wrong:

1. Division by zero. The denominator must not be zero.
2. The square root of a negative number. The number under an even root must be zero or more.

Example 1. $f(x) = \dfrac{1}{x - 3}$. The denominator is zero at $x = 3$, so the domain is $\mathbb{R} - \{3\}$, which is $(-\infty, 3) \cup (3, \infty)$.

Example 2. $g(x) = \sqrt{x - 2}$. We need $x - 2 \ge 0$, so $x \ge 2$ and the domain is $[2, \infty)$.

Example 3. $h(x) = \dfrac{\sqrt{x - 2}}{x - 5}$. Both rules apply: $x \ge 2$ and $x \neq 5$. The domain is $[2, 5) \cup (5, \infty)$.

Move $a$ and $b$ apart and the domain grows. Slide $a$ past $b$ and the curve disappears: no $x$ makes both roots real.

Finding the range

To find the range, we write $y = f(x)$ and ask: for which $y$ can we solve for $x$?

Example. $g(x) = \sqrt{9 - x^2}$. The domain needs $9 - x^2 \ge 0$, so $-3 \le x \le 3$. For these $x$, $x^2$ goes from $0$ to $9$, so $9 - x^2$ goes from $9$ down to $0$, and the root goes from $3$ down to $0$. The range is $[0, 3]$.

Example. $f(x) = \dfrac{1}{x - 3}$. Put $y = \dfrac{1}{x - 3}$. Then $x - 3 = \dfrac{1}{y}$, which works for every $y \neq 0$. So the range is $\mathbb{R} - \{0\}$.

When are two functions equal?

Two functions are equal when they have the same domain and the same value at every point. Take $f(x) = \dfrac{x^2 - 4}{x - 2}$ and $g(x) = x + 2$. For $x \neq 2$ they are the same number, but $f$ is not defined at $x = 2$ and $g$ is. The domains differ, so $f \neq g$.

Practise with new numbers

Try it in Python

This program tries many values of $x$ in the formula $\dfrac{\sqrt{x-2}}{x-5}$ and finds where it works.

Remember these

• A function gives exactly one output for each element of the domain.
• Vertical line test: no vertical line meets the graph twice.
• Domain: rule out division by zero and the even root of a negative number.
• Range: the values of $y$ that actually occur.
• Equal functions have the same domain and the same rule.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Is $\{(1, 2), (2, 3), (3, 3)\}$ a function? Is $\{(1, 2), (1, 3)\}$?

2. Find the domain of $f(x) = \sqrt{x - 7}$.

3. Find the domain of $f(x) = \dfrac{x}{x^2 - 16}$.

4. Find the range of $f(x) = x^2 + 4$.