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Complete notes using the video on the next assignment. Your teacher will give you a paper copy of the notes.

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Interactivate: Function Machine

Function Machine: Students investigate very simple functions by trying to guess the algebraic form from inputs and outputs. Function Machine is one of the Interactivate assessment explorers.

On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science. Additionally, the National Computational Science Institute (NCSI) provides nation-wide workshops portraying resources and instructional ideas to middle school, high school, and undergraduate instructors for use in the classroom. Resources and materials offered to these instructors are available free of charge from Shodor's website and are largely developed by Shodor student interns. Shodor's academic program efficiently guides participants from excitement to experience to expertise through computational explorations, research opportunities, and service.

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Interactivate: Function Machine On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science.

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Study this web site then do quiz.

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In order to understand the vertical line test, it's important that you understand what a function is. Make sure to check out the intro to functions lesson if you're new to the idea. You can think of a function like a vending machine. A function takes an input (like the code you enter on the vending machine) and gives you an output (like the candy or pop you get out of the vending machine).

An important rule that all functions must follow is that they can only assign ONE output to an input. For example, if you enter the code D4 in the vending machine and it drops a bag of pretzels, it must drop a bag of pretzels every time you enter the code D4. It can't give you a bag of pretzels sometimes and other times a bag of chips. Every input must be matched with exactly one output. It's ok if there's another input that gives the same output. For example, there might be two slots in the vending machine that both hold pretzels. So you may be able to enter the code D4 or D5 to get pretzels. It's ok if two different inputs give the same output. It's not ok if you enter the same input and get a different result.

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In order to understand the vertical line test, it's important that you understand what a function is. Make sure to check out the intro to functions lesson if you're new to the idea. You can think of a function like a vending machine.

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

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.

© Transum Mathematics 1997-2023
Scan the QR code below to visit the online version of this activity.

https://Transum.org/go/?Num=695

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.

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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?

TI-nSpire:

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

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