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GCSEMathsOCRRevision Notes6. AlgebraFunctionsComposite & Inverse Functions

Composite & Inverse Functions (OCR GCSE Maths)

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Mark

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Mark

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Composite Functions

What is a composite function?

  • Questions can involve more than one function
  • A composite function is one function applied to the output of another function
    • The input goes through the 1st function to become an output
    • This output goes through the 2nd function to become a new output
  • Composite functions may also be referred to as compound functions

Worked example

Two functions, A and B, are given by:

Function space straight A colon space space space space space space space space space input space rightwards arrow space box enclose space cross times 3 space end enclose space rightwards arrow space box enclose space minus 8 space end enclose space rightwards arrow space output space Function space straight B colon space space space space space space space space space input space rightwards arrow space box enclose space plus 5 space end enclose space rightwards arrow space box enclose space cross times 2 space end enclose space rightwards arrow space output

A composite function, C, is given by

Function space straight C colon space space space space space space space space input space rightwards arrow space box enclose Function space straight A end enclose space rightwards arrow space box enclose space Function space straight B space end enclose space rightwards arrow space output
 

(a)

If the input to function C is 7, find the output.
 

Substitute the number 7 into function A
 

7 × 3 - 8
 

Work out this value
 

13

 

Substitute 13 into function B
 

(13 + 5) × 2
 

Work out this value

36

the output of function C is 36

(b)
If the input to function C is x and the ouput is 4x, find the value of x.
 

Substitute the letter x into function A
 

x × 3 - 8
 

Simplify this expression
 

3x - 8
 

Substitute 3x - 8 into function B
 

(3x - 8 + 5) × 2
 

Simplify this expression
 

(3x - 3) × 2
= 6x - 6
 

The output is 6x - 6
The question says the output of function C is meant to be 4x
Set these two outputs equal to each other
 

6x - 6 = 4x
 

Solve this equation to find x
 

2x = 6
 x = 3

x = 3

Inverse Functions

What is an inverse function?

  • An inverse function reverses (undoes) the operations of a function
    •  Let's say a function “doubles the input, then adds 1”
      • Its inverse function must "subtract 1, then halve the result"
  • The inverse of space input space rightwards arrow space box enclose space cross times 6 space end enclose space rightwards arrow space box enclose space plus 4 space end enclose space rightwards arrow space output is space input space rightwards arrow space box enclose space minus 4 space end enclose space rightwards arrow space box enclose space divided by 6 space end enclose space rightwards arrow space output
    • The order of the boxes is reversed
    • The operation is replaced by its inverse operation
      • × ↔ ÷ and  + ↔ -
  • If a number goes through a function, then that result goes through the inverse function, you get back the same number again!

How can I use algebra to find inverse functions?

  • If putting x into a function gives out y, then putting y into the inverse function gives back x
  • Inverse functions are related to rearranging formulae
    • Let's say a function is space x space rightwards arrow space box enclose space cross times 2 space end enclose space rightwards arrow space box enclose space plus 3 space end enclose space rightwards arrow space y
      • It's inverse is therefore space x space rightwards arrow space box enclose space minus 3 space end enclose space rightwards arrow space box enclose space divided by 2 space end enclose space rightwards arrow space y
    • In algebra, the original function is 2 x plus 3 equals y
    • Look what happens when you make x the subject
      • x equals fraction numerator y minus 3 over denominator 2 end fraction
      • This is the function space y space rightwards arrow space box enclose space minus 3 space end enclose space rightwards arrow space box enclose space divided by 2 space end enclose space rightwards arrow space x 
      • The operations are identical to that of the inverse function! (the input here is a y though)
    • Swapping the x and y shows the inverse function more clearly
      • space x space rightwards arrow space box enclose space minus 3 space end enclose space rightwards arrow space box enclose space divided by 2 space end enclose space rightwards arrow space y

Worked example

A function is given by
 

space input space rightwards arrow space box enclose space cross times 4 space end enclose space rightwards arrow space box enclose space minus 5 space end enclose space rightwards arrow space output
 

(a)
If the output is 7, find the input.
 

Method 1
Reverse the order and operations to find the inverse function
 

space input space rightwards arrow space box enclose space plus 5 space end enclose space rightwards arrow space box enclose space divided by 4 space end enclose space rightwards arrow space output
 

Substitute 7 as the input into the inverse function
 

(7 + 5) ÷ 4
 

Work out this value

3

Method 2
If the output is 7, let the input to the function be x
 

table row cell x cross times 4 minus 5 end cell equals 7 row cell 4 x minus 5 end cell equals 7 end table
 

Solve this equation to find x
 

table row cell 4 x end cell equals 12 row x equals 3 end table

3

(b)
Find an algebraic expression for the inverse function, where the input is x.
 

Method 1
Reverse the order and operations to find the inverse function
 

space input space rightwards arrow space box enclose space plus 5 space end enclose space rightwards arrow space box enclose space divided by 4 space end enclose space rightwards arrow space output
 

Use x as the input
 

(x + 5) ÷ 4
 

Work out this value

the inverse function is fraction numerator bold italic x bold plus bold 5 over denominator bold 4 end fraction

Method 2
Write the original function in algebra (with x as the input and y as the output)
 

table row cell x cross times 4 minus 5 end cell equals y row cell 4 x minus 5 end cell equals y end table
 

Make x the subject of this equation
 

table row cell 4 x end cell equals cell y plus 5 end cell row x equals cell fraction numerator y plus 5 over denominator 4 end fraction end cell end table
 

This shows the "structure" of the inverse function, but currently uses a y
Replace the y with an x (the question wanted x as the input, not y)

the inverse function is fraction numerator bold italic x bold plus bold 5 over denominator bold 4 end fraction

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    Mark

    Author: Mark

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.