# Functions

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

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

This is level 5, simplify the composite functions. You can earn a trophy if you get at least 9 correct and you do this activity online.

Here are the functions to be used in the questions that follow.

$$f(x) = 3x+3$$,   $$g(x) = 2x^2$$   and   $$h(x) = 3 - 5x$$

 $$fg(9)$$ $$gh(8)$$ $$hg(2)$$ $$gf(4)$$ $$fgh(2)$$ $$ff(8)$$ $$fg(x)$$ $$gh(x)$$ $$hg(x)$$ $$fh(x)$$ $$fgh(x)$$ $$hh(x)$$
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This is Functions level 5. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 6

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

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Inverse Joke, Anon

Tuesday, June 16, 2020

"I poured root beer into a square glass. Now I just have beer!"

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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.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

## 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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