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

Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Exam-Style Description Help More 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.

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

Correct Wrong

\(x \to \)\( \times 4 \)\( \to \)\( + 1 \)\( \to \)

Correct Wrong

\(x \to \)\( \times 2 \)\( \to \)\( - 8 \)\( \to \)

Correct Wrong

\(x \to \)\( - 7 \)\( \to \)\( \times 3 \)\( \to \)

Correct Wrong

\(x \to \)\( + 2 \)\( \to \)\( \times 6 \)\( \to \)

Correct Wrong

\(x \to \)\( + 8 \)\( \to \)\( \times 3 \)\( \to \)

Correct Wrong

\(x \to \)\( \times 2 \)\( \to \)\( + 8 \)\( \to \)

Correct Wrong

\(x \to \)\( \times 7 \)\( \to \)\( - 4 \)\( \to \)

Correct Wrong

\(x \to \)\( - 7 \)\( \to \)\( \times 13 \)\( \to \)

Correct Wrong

\(x \to \)\( + 12 \)\( \to \)\( \times 13 \)\( \to \)

Correct Wrong

\(x \to \)\( + 17 \)\( \to \)\( \times 26 \)\( \to \)

Correct Wrong

\(x \to \)\( \times 56 \)\( \to \)\( - 7.5 \)\( \to \)

Correct Wrong


This is Functions level 1. You can also try:
Level 2 Level 3 Level 4 Level 5 Level 6


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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Tuesday, June 16, 2020

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

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Description of Levels



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.

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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 \(f(x)\).


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

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.


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.


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


\(x+7 = 10\)

\(x = 3\)

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.


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

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

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

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

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.


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 \( fg(3) = 97\)


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.


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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