# Functions

## An online exercise on function notation, inverse functions and composite functions.

##### Level 1Level 2Level 3Level 4Level 5Level 6Exam-StyleDescriptionHelpMore Algebra

This is level 1, describe function machines using function notation. You can earn a trophy if you get at least 9 correct and you do this activity online. The first question has been done for you.

 $$x \to$$$$\times 3$$$$\to$$$$+ 4$$$$\to$$ $$x \to$$$$\times 7$$$$\to$$$$+ 3$$$$\to$$ $$x \to$$$$\times 2$$$$\to$$$$- 8$$$$\to$$ $$x \to$$$$- 9$$$$\to$$$$\times 4$$$$\to$$ $$x \to$$$$+ 2$$$$\to$$$$\times 7$$$$\to$$ $$x \to$$$$+ 8$$$$\to$$$$\times 2$$$$\to$$ $$x \to$$$$\times 3$$$$\to$$$$+ 7$$$$\to$$ $$x \to$$$$\times 6$$$$\to$$$$- 4$$$$\to$$ $$x \to$$$$- 9$$$$\to$$$$\times 12$$$$\to$$ $$x \to$$$$+ 14$$$$\to$$$$\times 20$$$$\to$$ $$x \to$$$$+ 19$$$$\to$$$$\times 25$$$$\to$$ $$x \to$$$$\times 43$$$$\to$$$$- 9.5$$$$\to$$
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This is Functions level 1. You can also try:
Level 2 Level 3 Level 4 Level 5 Level 6

## Instructions

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## Description of Levels

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Level 1 - Describe function machines using function notation.

Level 2 - Evaluate the given functions.

Level 3 - Solve the equations given in function notation.

Level 4 - Find the inverse of the given functions.

Level 5 - Simplify the composite functions.

Level 6 - Mixed questions.

Exam Style questions are in the style of GCSE or IB/A-level exam paper questions and worked solutions are available for Transum subscribers.

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## Functions

The following notes are intended to be a reminder or revision of the concepts and are not intended to be a substitute for a teacher or good textbook.

### Level 1: Describe function machines using function notation.

Function notation is quite different to the algebraic notation you have learnt involving brackets. $$f(x)$$ does not mean the value of f multiplied by the value of x. In this case f is the name of the function and you would read $$f(x) = x^2$$ as "f of x equals x squared".

In terms of function machines, if the input is $$x$$ then the output is $$f(x)$$.

Example

$$x \to$$$$+ 3$$$$\to$$$$\times 4$$$$\to f(x)$$

In this case 3 is added to $$x$$ and then the result is multiplied by 4 to give $$f(x)$$

$$(x+3) \times 4 = f(x)$$

$$f(x) = 4(x+3)$$

### Level 2: Evaluate the given functions.

Example

if $$f(x)=x^2 + 3$$ calculate the value of $$f(6)$$

This means replace the $$x$$ with a 6 in the given function to obtain the result.

$$f(6) = 6^2+3$$

$$f(6) = 39$$

### Level 3: Solve the equations given in function notation.

Example

$$f(x)=3(x+7)$$ find $$x$$ if $$f(x) = 30$$

$$3(x+7)=30$$

$$x+7 = 10$$

$$x = 3$$

### Level 4: Find the inverse of the given functions.

The inverse of a function, written as $$f^{-1}(x)$$ can be thought of as a way to 'undo' the function. If the function is written as a function machine, the inverse can be thought of as working backwards with the output becomming the input and the input becoming the output.

Example

$$f(x) = 4(x+3)$$

$$x \to$$$$+ 3$$$$\to$$$$\times 4$$$$\to f(x)$$

$$f^{-1}(x) \leftarrow$$$$- 3$$$$\leftarrow$$$$\div 4$$$$\leftarrow x$$

$$f^{-1}(x) = \frac{x}{4} - 3$$

A quicker way of finding the inverse of $$f(x)$$ is to replace the $$f(x)$$ with $$x$$ on the left side of the equals sign and replace the $$x$$ with $$f^{-1}(x)$$ on the right side of the equals sign. Then rearrange the equation to make $$f^{-1}(x)$$ the subject.

### Level 5: Simplify the composite functions.

A composite function contains two functions combined into a single function. One function is applied to the result of the other function. You should evaluate the function closest to $$x$$ first.

Example

if $$f(x)=2x+7$$ and $$g(x)=5x^2$$ find $$fg(3)$$

$$g(3) = 5 \times 3^2$$

$$g(3) = 5 \times 9$$

$$g(3) = 45$$

$$f(45) = 2 \times 45 + 7$$

$$f(45) = 97$$

so $$fg(3) = 97$$

Example

if $$f(x)=x+2$$ and $$g(x)=3x^2$$ find $$gf(x)$$

$$gf(x) = 3(x+2)^2$$

$$gf(x) = 3(x^2+4x+4)$$

$$gf(x) = 3x^2+12x+12$$

### Level 6: Mixed questions.

Example

Find $$f(x-2)$$ if $$f(x)=5x^2+3$$

$$f(x-2) =5(x-2)^2+3$$

$$f(x-2) =5(x^2-4x+4)+3$$

$$f(x-2) =5x^2-20x+20+3$$

$$f(x-2) =5x^2-20x+23$$

Did you know some calculators can apply functions?

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