# Functions: roots, composition and inverses · Year 11

> Explanation, worked examples and free practice for the national curriculum in England.

In Functions: roots, composition and inverses (Year 11) you practise 2 skills: Composing functions · The inverse function.

A root of a function is a value of x where the function is 0. When a polynomial is written as a product of factors, each factor gives a root. A squared factor gives a repeated root: there the graph touches the axis without crossing it. A polynomial of odd degree always has at least one root, because its two ends have opposite signs.

In a composition you put one function inside another and work from the inside out. The inverse function works the other way: from each value it gives back the x it came from. A function has an inverse only when each value comes from one x alone, so sometimes the domain is cut down, as with a parabola where only x of 0 or more is kept.

To solve an inequality, find the roots and test the sign at one point in each interval between them. A limit describes where a function is heading: a fixed number divided by a number that grows without bound gets closer to 0, and a fixed number divided by a number that gets closer to 0 grows without bound in size.

**Tip:** Before you choose an answer, put one number in and check it: it is the quickest way to catch a sign mistake.

## What you learn

### Composing functions

- Example: Here are the functions f(x) = 3x − 4 and g(x) = x² + 3x + 2 . Find f(g(−2)) .
- Answer: −4
- Solution: Inside: g(−2) = (−2)² + 3 × (−2) + 2 = 0 . Then f(0) = 3 × 0 − 4 = −4 .

### The inverse function

- Example: Find the inverse function of f(x) = 2x + 9 .
- Answer: 0.5x − 4.5
- Solution: Write y = 2x + 9 and swap the variables: x = 2y + 9 . Make y the subject: y = (x − 9) / 2 . So f⁻¹(x) = 0.5x − 4.5 .

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

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