FunctionsAn online exercise on function notation, inverse functions and composite functions. 
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 \(h(x) = 1  4x\)
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www.transum.org/Maths/Exercise/Functions.asp?Level=5
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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/Alevel 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 \(f(x2)\) if \(f(x)=5x^2+3\)
\(f(x2) =5(x2)^2+3\)
\(f(x2) =5(x^24x+4)+3\)
\(f(x2) =5x^220x+20+3\)
\(f(x2) =5x^220x+23\)
TInSpire:
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