Composite & Inverse Functions (Cambridge O Level Maths)

What is a composite function?

• A composite function is one function applied to the output of another function
• Composite functions may also 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 
    • 
    • 

• 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 
    • 
    • 

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 4
    Rewrite using the correct notation for an inverse function
    
    • 

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  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:

How do I find the domain and range of an inverse function?

• The domain of an inverse function is the same as the range of the original function
• The range of an inverse function is the same as the domain of the original function