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Composite & Inverse Functions (AQA GCSE Maths)
Revision Note
Composite Functions
What is a composite function?
- A composite function is one function applied to the output of another function
- Composite functions may also be referred to as compound functions
What do composite functions look like?
- The notation you will see for a composite function is fg(x)
- This can be written as f(g(x)) and means “f applied to the output of g(x)”
- i.e. g(x) happens first
- Always apply the function on the outside to the output of the function on the inside
- gf(x) means g(f(x)) and means “g applied to the output of f(x)”
- i.e. f(x) happens first
How does a composite function work?
- If you are putting a number into fg(x)
- STEP 1
Put the number into g(x) - STEP 2
Put the output of g(x) into f(x) - For example, if and
- STEP 1
- If you are using algebra, to find an expression for a composite function
- STEP 1
For fg(x) put g(x) wherever you see x in f(x) - STEP 2
Simplify if necessary - For example, if and
- STEP 1
Examiner Tip
- Make sure you are applying the functions in the correct order
- The letter nearest the bracket is the function applied first
Worked example
In this question, and .
(a)
Find .
g is on the inside of the composite function so apply g first.
Apply f to the output of g.
(b)
Find .
f is on the inside of the composite function so apply f first by substituting the function f(x) into g(x).
Simplify.
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Inverse Functions
What is an inverse function?
- An inverse function does the exact opposite of the function it came from
- For example, if the function “doubles the number and adds 1” then its inverse is
- “subtract 1 and halve the result”
- It is the inverse operations in the reverse order
How do I write inverse functions?
- An inverse function f-1 can be written as or
- For example, if its inverse can be written as
- or
How do I find an inverse function?
- The easiest way to find an inverse function is to 'cheat' and swap the and variables
- Note that this is a useful method but you MUST remember not to do this in any other circumstances in maths
- STEP 1
Write the function in the form - STEP 2
Swap the s and s to get - STEP 3
Rearrange the expression to make the subject again - STEP 4
Write as f-1(x) = … (or f-1 : x ↦ …)
- should not exist in the final answer
- For example, if its inverse can be found as follows
- STEP 1
Write the function in the form - STEP 2
Swap the and to get - STEP 3
Rearrange the expression to make the subject again
- STEP 1
-
- STEP 4
Rewrite using the correct notation for an inverse function
- STEP 4
How does a function relate to its inverse?
- If then the input of 3 gives an output of 10
- The inverse function undoes f(x)
- An input of 10 into the inverse function gives an output of 3
- If then
-
- If you apply a function to x, then immediately apply its inverse function, you get x
- Whatever happened to x gets undone
- f and f-1 cancel each other out when applied together
- If you apply a function to x, then immediately apply its inverse function, you get x
- If and you want to solve
- Finding the inverse function in this case is tricky (impossible if you haven't studied logarithms)
- instead, take f of both sides and use that cancel each other out:
Worked example
Find the inverse of the function .
Write the function in the form and then swap the and
Rearrange the expression to make the subject again.
Rewrite using the correct notation for an inverse function.
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