Composite & Inverse Functions | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Composite Functions

What are Composite Functions?

• A composite function involves applying one function to the output of another function. In simpler terms, it's like performing one operation after another.
• Another term for composite functions is compound functions.

What do composite functions look like?

• Composite function notation is written as fg(x). 
  • This can be written as f(g(x)) and means “ f applied to the output of g(x)”. 
  • In this case, g(x) happens first. 
• Always apply the outer function to the output of the inner function. 
  • gf(x) means g(f(x)) and means “g applied to the output of f(x)”. 
  • In this case, 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 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
  • 
  • 

Inverse Functions

• An inverse function performs the opposite operation of the original function. For instance, if the initial function "doubles the number and adds 1," the inverse function would "subtract 1 and halve the result." It essentially reverses the operations in the opposite order.
• Inverse functions essentially undo the actions of the original function.

How do I write inverse functions?

• An inverse function f^-1 can be written as
  • For example, if f(x) = 2x + 1 its inverse can be written as
  • 

How do I find an inverse function?

• The easiest way to find an inverse function is to 'cheat' and swap the x and y variables.
• Note: This is useful here but should not be done in other math contexts.
• Step-by-Step Process:
  • STEP 1: Write the function in the form y = …
  • STEP 2: Swap the 𝑥x and 𝑦y to get x = …
  • STEP 3: Rearrange the expression to make 𝑦y the subject again
  • STEP 4: Write as f⁻¹(x) = … (or f−1 : x↦…)
    • Ensure 𝑦y does not exist in the final answer
• Example:
  • Given f(x) = 2x + 1, find its inverse:
  • STEP 1: Write the function as y = 2x + 1
  • STEP 2: Swap x and y: x = 2y + 1
  • STEP 3: Rearrange to make y the subject:
    
  • STEP 4: Write the inverse function:
    

How does a function relate to its inverse?

• If f(3) = 10, 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. 
  • Therefore, if f(3) = 10, then f⁻¹ (10) = 3. 
• The relationship between a function and its inverse is: 
  • f(f ⁻¹ (x)) = f ⁻¹ (f(x)) = x 
  • Applying a function to x, then immediately applying its inverse function, returns x. 
  • f and f⁻¹ cancel each other out when applied together. 
• If f(x) = 2x and you want to solve f⁻¹ (x) = 5: 
  • Finding the inverse function f −1 (x) directly is difficult without knowledge of logarithms. 
  • Instead, take f of both sides and use the fact that f and f⁻¹ 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