Functions Toolkit

Introduction to Functions

What is a function?

A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers
- It may be thought of as a mathematical “machine”
- For example, if the function (rule) is “double the number and add 1”, the two mathematical operations are "multiply by 2 (×2)" and "add 1 (+1)"
  - Putting 3 in to the function would give 2 × 3 + 1 = 7
  - Putting -4 in would give 2 × (-4) + 1 = -7
- Putting $x$ in would give $2 x + 1$
The number being put into the function is often called the input
The number coming out of the function is often called the output

What does a function look like?

A function f can be written as f(x) = … or f : x ↦ …
- These two different types of notation mean exactly the same thing
- Other letters can be used. g, h and j are common but any letter can technically be used
  - Normally, a new letter will be used to define a new function in a question
For example, the function with the rule “triple the number and subtract 4” would be written
- $f (x) = 3 x - 4$ or $f : x \mapsto 3 x - 4$
In such cases, $x$ would be the input and $f (x)$ would be the output
Sometimes functions don’t have names like f and are just written as y = …
- eg. $y = 3 x - 4$

How does a function work?

A function has an input $(x)$ and output $(f (x) or y)$
Whatever goes in the bracket (instead of $x$ )with f, replaces the $x$ on the other side
- This is the input
If the input is known, the output can be calculated
- For example, given the function $f (x) = 2 x + 1$
  - $f (3) = 2 \times 3 + 1 = 7$
  - $f (- 4) = 2 \times (- 4) + 1 = - 7$
  - $f (a) = 2 a + 1$

If the output is known, an equation can be formed and solved to find the input
- For example, given the function $f (x) = 2 x + 1$
  - If $f (x) = 15$ , the equation $2 x + 1 = 15$ can be formed
  - Solving this equation gives an input of 7

Worked example

A function is defined as $f (x) = 3 x^{2} - 2 x + 1$ .

(a)

Find

f (7)

The input is

x = 7

, so substitute 7 into the expression everywhere you see an

x

f (7) = 3 {(7)}^{2} - 2 (7) + 1

Calculate.

\begin{array}{rcl} f (7) & = & 3 (49) - 14 + 1 \\ = & 147 - 14 + 1 \end{array}

f (7) = 134

(b)

Find

f (x + 3)

The input is

x = x + 3

so substitute

x + 3

into the expression everywhere you see an

x

$f (x + 3) = 3 {(x + 3)}^{2} - 2 (x + 3) + 1$

Expand the brackets and simplify.

$\begin{array}{rcl} f (x + 3) & = & 3 (x^{2} + 6 x + 9) - 2 (x + 3) + 1 \\ = & 3 x^{2} + 18 x + 27 - 2 x - 6 + 1 \\ = & 3 x^{2} + 16 x + 22 \end{array}$

$f (x + 3) = 3 x^{2} + 16 x + 22$

A second function is defined $g : x \mapsto 3 x - 4$ .

(c)

Find the value of