- 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 in would give

- The number being put into the function is often called the
**input** - The number coming out of the function is often called the
**output**

- 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
- or

- In such cases, would be the input and would be the output
- Sometimes functions don’t have names like f and are just written as y = …
- eg.

- A function has an
**input**and**output** - Whatever goes in the bracket (instead of )with f, replaces the on the other side
- This is the input

- If the input is known, the output can be calculated
- For example, given the function

- For example, given the function

- If the output is known, an equation can be formed and solved to find the input
- For example, given the function

- If , the equation can be formed
- Solving this equation gives an input of 7

- For example, given the function

A function is defined as .

(a)

Find .

The input is , so substitute 7 into the expression everywhere you see an .

Calculate.

(b)

Find .

The input is so substitute into the expression everywhere you see an .

Expand the brackets and simplify.

A second function is defined .

(c)

Find the value of for which .

Form an equation by setting the function equal to -16.

Solve the equation by first adding 4 to both sides, then dividing by 3.

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