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Here we will learn about functions in algebra, including function machines, composite functions and inverse functions.
There are also functions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
Functions in algebra are used to describe the operation being applied to an input in order to get an output.
Functions can be presented using a function machine.
Step-by-step guide: Function machines
Functions can be described using function notation.
The f in f(x) is known as the function that being applied to a variable x . Other letters such as g and h are also commonly used.
E.g.
Let’s look at a function
Then let’s rewrite this using function notation.
Step-by-step guide: Function notation
Functions can be extended to work with inverse functions and composite functions. Inverse functions ‘reverse’ or ‘undo’ the original function. Composite functions combine two or more functions.
We can combine functions to make composite functions.
Step-by-step guide: Composite functions
In order to solve an equation using a function machine:
Get your free functions in algebra worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEGet your free functions in algebra worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREESolve \frac{x}{3}+7=3
The order of operations is \div 3 , then +7 .
2Draw a function machine starting with the unknown as the input and the value of the equation as the output.
3Work backwards, applying inverse operations to find the unknown input.
x=\left( 3-7 \right)\times 3 x=-12In order to evaluate a function using function notation:
Find f(2) when f(x) = 3x + 2
Write out the function for x using function notation, replacing the x with an empty set of brackets.
Replace the x in the function with the number or algebraic term in the brackets next to the “name” of the function.
Apply the correct operations to the number or term as appropriate and simplify.
Find f(4m) when f(x) = 5x − 7
Write out the function for x using function notation, replacing the x with an empty set of brackets.
Replace the x in the function with the number or algebraic term in the brackets next to the “name” of the function.
Apply the correct operations to the number or term as appropriate and simplify.
In order to evaluate composite functions:
If f(x) = 3x and g(x) = x^2 + 1 , find fg(2) :
Use the number to be evaluated as the input for the inner function and substitute it into the expression.
Find the output for the inner function and substitute it into the expression for the outer function.
Find the output for the outer function and repeat the process if there are any further outer functions.
f(5) = 15 , so fg(2) = 15 .
In order to find composite functions:
If f(x) = 2x - 1 and g(x) = x^2 + 4 , find fg(x) :
Take the most inner function and substitute it into the next outer function wherever there is an x .
Simplify the expression as appropriate.
Repeat for any further outer functions.
No further outer functions, so fg(x)=2x^2+3 .
In order to find an inverse function.
If f(x) = 3x + 7 , find f^{-1}(x) .
Write out the expression for the original function using a y instead of the x . Set this expression equal to x .
Rearrange the equation to make y the subject.
Write your inverse function using the f^{-1} notation.
When using function machines that include multiple operations to solve equations, a common error is to forget to work backwards. This means that the inverse operations are used but they are in the wrong order.
A common error is to not follow the correct order of operations when creating a function machine for an equation.
It is common for f(x) to be thought of as “f times x” rather than “f of x ”. This confusion can lead to incorrect evaluations of values.
E.g.
When finding f(2) when f(x)=x+3 , a student may think f(2) means 2(x+3) .
The correct solution is
\begin{aligned} &f(x)=x+3 \\\\ &f(2)=2+3 \\\\ &f(2)=5 \end{aligned}A common mistake is to think that fg(x) means f(x) \times g(x) .
E.g.
If f(x) = x + 1 and g(x) = 2x - 3 , a common error is be to think that
fg(x) = (x + 1)(2x - 3) , rather than the correct answer of fg(x) = 2x - 2.
A common error is to apply the functions in the wrong order.
fgh(x) means we should apply the functions in the order h , then g , then f , which is right to left.
Thinking of fgh(x) as f[g(h(x))] can be help. h is the innermost function and we need to work outwards.
A common error is to think the inverse function will be the reciprocal of the function because of the “power of -1 ” notation.
E.g.
If f(x)=3x , an incorrect expression for f^{-1} would be (3x)^{-1} or \frac{1}{3x}.
1. Find the missing Output and missing Input for the function machine.
Work forwards to find a and then backwards using inverse operations to find b .
2. Select the correct function machine and solution for the equation:
2x+6=10
x is the input. The first operation tells us to multiply by 2 and the second operation tells us to add 6 .
This gives an output of 10. when x = 2 .
3. Find f(7) when f(x)=2x-8
Replace the x with 7 and simplify.
\begin{aligned} f(7)) &= 2(7) – 8 \\ &= 14 – 8\\ &= 6 \end{aligned}4. Find g(3k+2) when g(x)=6x-5
Replace the x with 3k+2, expand the brackets and simplify.
\begin{aligned} g(3k+2) &= 6(3k+2) – 5 \\ &= 18k + 12 -5\\ &= 18k +7 \end{aligned}5. Find fg(x) when f(x)=x^2+1 and g(x)=2x-3
6. If g(x)=\frac{x}{2}-7 , find g^{-1}(x):
Rearranging the formula, first add 7 , then multiply by 2.
1.
(a) Find the output if the input is 10.
(b) Write a simplified expression for the input if the output is 24n-32
(4 marks)
(a)
56
(1)
(b)
\frac{24n-32}{8}
(1)
3n-4+3
(1)
3n-1
(1)
2. If f(x)=5-2x and g(x)=2x+6
(a) Find x when f(x)=-9
(b) Write an expression for gf(x)
(5 marks)
(a)
Sight of 2x=14
(1)
x=7
(1)
(b)
2(5-2x)+6
(1)
10-4x+6
(1)
16-4x
(1)
3. g(x)=\frac{x-3}{4} and h(x)=\frac{x}{2}+3
Solve when g^{-1}(x)=h^{-1}(x)
(4 marks)
(1)
h^{-1}(x)=2(x-3) or 2x-6
(1)
Set equal 4x+3=2x-6
(1)
Answer x=-4.5
(1)
You have now learned how to:
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8 sets of free exam practice papers written by maths teachers and examiners for Edexcel, AQA and OCR.
Each set of exam papers contains the three papers that your students will expect to find in their GCSE mathematics exam.