FunctionsAn online exercise on function notation, inverse functions and composite functions. |
This is level 6, mixed questions. You can earn a trophy if you get at least 9 correct and you do this activity online.
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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.
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.
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 \(f(x)\)
\( (x+3) \times 4 = f(x) \)
\( f(x) = 4(x+3) \)
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.
\(f(6) = 6^2+3\)
\(f(6) = 39\)
Example
\(f(x)=3(x+7) \) find \(x\) if \(f(x) = 30\)
\(3(x+7)=30\)
\(x+7 = 10\)
\(x = 3\)
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.
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 \(gf(x)\)
\( gf(x) = 3(x+2)^2\)
\( gf(x) = 3(x^2+4x+4) \)
\( gf(x) = 3x^2+12x+12 \)
Example
Find \(f(x-2)\) if \(f(x)=5x^2+3\)
\(f(x-2) =5(x-2)^2+3\)
\(f(x-2) =5(x^2-4x+4)+3\)
\(f(x-2) =5x^2-20x+20+3\)
\(f(x-2) =5x^2-20x+23\)
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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Inverse Joke, Anon
Tuesday, June 16, 2020
"I poured root beer into a square glass. Now I just have beer!"