Functions and function notation

What a function is, independent and dependent variables, natural domains, and how to use f(x) notation, with interactive examples and practice.

What is a function?

Think of a function as a machine. You put a number in, the machine follows its rule, and exactly one number comes out. The rule "square the number, then add 1" is a function: put in $3$ and you always get $10$.

We write this rule as $y = x^2 + 1$. The input $x$ is the , and the output $y$ is the , because its value depends on what you put in.

A table of values shows a function at work. For $y = x^2 + 1$:

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$5$	$2$	$1$	$2$	$5$	$10$

Different inputs can give the same output ($-2$ and $2$ both give $5$). That is fine. What a function never does is give two different outputs for one input.

Function notation

Instead of $y = x^2 + 1$ we often write $f(x) = x^2 + 1$. Read $f(x)$ as "f of x". The letter $f$ is the name of the function, and the bracket holds the input.

So $f(3)$ means "the output when the input is $3$": $f(3) = 3^2 + 1 = 10$.

Function notation

$$y = f(x) \quad\text{means}\quad \text{output} = f(\text{input})$$

Feed the machine

Drag the orange point along the curve. The readout shows the input and the one output the function gives.

Try this

Find the two inputs that give an output of $5$.
Is there any input that gives an output of $0$? Why not?

Example 1●

If $f(x) = x^2 - 3x$, find $f(4)$, $f(-2)$ and $f(0)$.

$$f(4) = (4)^2 - 3(4) = 16 - 12 = 4$$

Replace every x with the number, in brackets so the signs stay correct.

$$f(-2) = (-2)^2 - 3(-2) = 4 + 6 = 10$$

The brackets matter most with negatives: (−2) squared is +4, and −3 times −2 is +6.

$$f(0) = 0^2 - 3(0) = 0$$

Substituting expressions

The input can be an expression, not just a number. Whatever is in the bracket replaces every $x$.

Example 2●●

If $f(x) = x^2 - 3x$, find and simplify $f(a + 1)$.

$$f(a+1) = (a+1)^2 - 3(a+1)$$

Swap every x for the whole expression a + 1, kept in brackets.

$$= a^2 + 2a + 1 - 3a - 3$$

Expand (a + 1)² = a² + 2a + 1, and −3(a + 1) = −3a − 3.

$$= a^2 - a - 2$$

Collect like terms.

Natural domain: what inputs are allowed?

The is the set of inputs a function can take. When no domain is given, we use the natural domain: every real number that gives a real output. Only two things usually cause trouble in Year 11:

Dividing by zero. In $f(x) = \dfrac{1}{x - 2}$, $x$ cannot be $2$.
Square roots of negatives. In $g(x) = \sqrt{x + 3}$, we need $x + 3 \ge 0$, so $x \ge -3$.

Example 3●●

State the natural domain of $h(x) = \dfrac{5}{\sqrt{4 - x}}$.

$4 - x \ge 0$

A square root needs what is inside it to be zero or more.

$4 - x > 0$

But this root is also in the denominator, so it cannot be zero either.

$x < 4$

Solve: add x to both sides. The domain is all real x less than 4.

Flashcards

Function

A rule that gives exactly one output for each allowed input.

Independent variable

The input, usually $x$. You choose it freely.

Dependent variable

The output, usually $y$. Its value depends on the input.

Natural domain

All real inputs that give a real output. Watch for division by zero and square roots of negatives.

Is $f(a+1)$ the same as $f(a)+1$?

No. Substitute the whole bracket $a+1$ for every $x$.

Practice

If $f(x) = 2x^2 - 1$, what is $f(-3)$?
Which statement about a function is true?
What is the natural domain of $g(x) = \sqrt{x - 5}$?
If $f(x) = 3x - 2$ and $f(a) = 19$, find $a$.
If $f(x) = x^2$, which expression equals $f(x + h)$?
If $g(x) = 5 - 2x$, find $g(-3)$.
If $f(x) = x^2$, then $f(2a) =$
The natural domain of $f(x) = \dfrac{1}{x + 4}$ is: