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Inverse Functions (Cambridge (CIE) O Level Additional Maths): Revision Note

Exam code: 4037

Paul

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Paul

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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 space straight f to the power of negative 1 end exponent left parenthesis x right parenthesis space equals space horizontal ellipsis space space or  straight f to the power of negative 1 end exponent space colon space x space rightwards arrow from bar space horizontal ellipsis

    • For example, if straight f left parenthesis x right parenthesis space equals space 2 x space plus space 1 its inverse can be written as

    • straight f to the power of negative 1 end exponent left parenthesis x right parenthesis space equals space fraction numerator left parenthesis x space – space 1 right parenthesis space over denominator 2 end fraction  or   straight f to the power of negative 1 end exponent colon space x space rightwards arrow from bar space fraction numerator left parenthesis x space – space 1 right parenthesis over denominator 2 end fraction

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 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 formspace y space equals space horizontal ellipsis e.g.   y space equals space 2 x space plus space 1

  • STEP 2 Swap the x's andspace y's to get x space equals space horizontal ellipsis e.g.  x space equals space 2 y space plus space 1

  • STEP 3 Rearrange the expression to make y the subject again table row cell x space minus space 1 space end cell equals cell space 2 y end cell row cell fraction numerator x space minus space 1 over denominator 2 end fraction space end cell equals cell space y space space space space space space rightwards arrow space space space y space equals space fraction numerator x space minus space 1 over denominator 2 end fraction end cell end table

  • STEP 4 Rewrite using the correct notation for an inverse function

    • either as f-1(x) = … or f-1 : x ↦ …

    • yshould not exist in the final answer

      • e.g.  straight f to the power of negative 1 end exponent open parentheses x close parentheses space equals space fraction numerator space x italic space minus space 1 over denominator 2 end fraction

How does a function relate to its inverse?

  • If straight f open parentheses 3 close parentheses equals 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

      • If straight f open parentheses 3 close parentheses equals 10 then straight f to the power of negative 1 end exponent open parentheses 10 close parentheses equals 3

  • ff to the power of negative 1 end exponent open parentheses x close parentheses equals straight f to the power of negative 1 end exponent straight f open parentheses x close parentheses equals x

    • 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 straight f open parentheses x close parentheses space equals space 2 to the power of x and you want to solve straight f to the power of negative 1 end exponent open parentheses x close parentheses space equals space 5

    • Finding the inverse function straight f to the power of negative 1 end exponent open parentheses x close parentheses in this case is tricky (impossible if you haven't studied logarithms)

    • instead, take f of both sides and use that ff to the power of negative 1 end exponent cancel each other out:

table row cell ff to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell straight f open parentheses 5 close parentheses end cell row x equals cell straight f open parentheses 5 close parentheses end cell row x equals cell 2 to the power of 5 equals 32 end cell end table

What condition is needed for an inverse function to exist?

  • For the inverse function to exist, straight f to the power of negative 1 end exponent open parentheses x close parentheses, the original function straight f open parentheses x close parentheses must be one-to-one

    • Substituting 1 input into straight f must give 1 output only

    • Substituting this 1 output into straight f to the power of negative 1 end exponent must give back the original input only

      • At no point are more values allowed to be created!

Worked Example

Find the inverse of the function straight f open parentheses x close parentheses space equals space 5 space minus space 3 x.

Write the function in the form y space equals space 5 space minus space 3 x and then swap the x and y.  

y space equals space 5 space minus space 3 x x space equals space 5 space minus space 3 y

Rearrange the expression to make y the subject again.

table row cell x space end cell equals cell space 5 space minus space 3 y end cell row cell space x space plus space 3 y space end cell equals cell space 5 end cell row cell 3 y space end cell equals cell space 5 space minus space x end cell row cell y space end cell equals cell space fraction numerator 5 space minus space x over denominator 3 end fraction end cell end table

  Rewrite using the correct notation for an inverse function.

Domain & range of inverse functions

How do I find the domain and range of inverse functions?

Domain and range of a function swap for its inverse

 

  • The range of a function will be the domain of its inverse function

  • The domain of a function will be the range of its inverse function

Worked Example

A function is defined as straight f stretchy left parenthesis x stretchy right parenthesis equals square root of 3 x minus 2 end root comma space space x greater than fraction numerator 3 over denominator 2 space end fraction.

Write down the domain and range of straight f to the power of negative 1 end exponent stretchy left parenthesis x stretchy right parenthesis.

The domain of an inverse function is the range of the function.

The range of straight f open parentheses x close parentheses is

straight f greater than 0

therefore The domain of  is bold italic x bold greater than bold 0

The range of an inverse function is the domain of the function.

therefore The range of is bold f to the power of bold minus bold 1 end exponent bold greater than bold 3 over bold 2

Graphs of inverse functions

  • The graph of an inverse function, y equals straight f to the power of negative 1 end exponent open parentheses x close parentheses, is a reflection of the graph of the function, y equals straight f open parentheses x close parentheses, in the line y equals x

  • Key features of the graph of y equals straight f open parentheses x close parentheses will be reflected, such as

    • x and y axes intercepts

    • turning points

    • asymptotes

How do I sketch the graph of an inverse function?

  • STEP 1

    • Sketch the line y equals x, and if need be, the graph of y equals straight f open parentheses x close parentheses

  • STEP 2

    • Reflect the graph of y equals straight f open parentheses x close parentheses in the line y equals x

      • Remember it is a sketch, but the graphs together should look like reflections

    • Consider points where the reflected graph will intersect the x and y axes

      • e.g.  The point open parentheses 4 comma space 0 close parentheses will be reflected to the point open parentheses 0 comma space 4 close parentheses

    • Consider any asymptotes on the graph of y equals straight f open parentheses x close parentheses - these will also be need reflecting

      • e.g.  The asymptote (line) x equals negative 2 will be reflected to the line y equals negative 2

    • Consider any restrictions on the domain and range of straight f open parentheses x close parentheses

      • e.g.  If the domain is x greater than 0 only sketch the graph for positive values of x

  • STEP 3

    • Label key points on the sketch of y equals straight f to the power of negative 1 end exponent open parentheses x close parentheses and state the equations of any asymptotes

  • This process works the other way round - the graph of y equals straight f open parentheses x close parentheses can be sketched from the graph of y equals straight f to the power of negative 1 end exponent open parentheses x close parentheses

Examiner Tips and Tricks

  • If not given, sketch the graphs of y equals straight f open parentheses x close parentheses and y equals x to help sketch the graph of the inverse, y equals straight f to the power of negative 1 end exponent open parentheses x close parentheses

  • If the graph of y equals straight f open parentheses x close parentheses is given you do not need to know the expression for straight f open parentheses x close parentheses to sketch y equals straight f to the power of negative 1 end exponent open parentheses x close parentheses

    • Just reflect whatever is given in the line y equals x

Worked Example

The diagram below shows the graph of y equals straight f open parentheses x close parentheses, where straight f open parentheses x close parentheses equals 4 minus 4 over x comma space space x greater than 0.

desmos-graph-6

a)On a copy of the diagram, sketch the graph of y equals straight f to the power of negative 1 end exponent open parentheses x close parentheses. Label the point where the graph crosses the y-axis and write down the equation of the asymptote.

The graph of an inverse function is the reflection of the graph of that function in the line y equals x.

Draw the line y equals x to help sketch the inverse function.

The x-axis intercept open parentheses 1 comma space 0 close parentheses becomes the y-axis intercept, open parentheses 0 comma space 1 close parentheses.

The (horizontal) asymptote y equals 4 will. become the (vertical) asymptote x equals 4.

desmos-graph-5

b) Use your sketch, or otherwise, to write down the value of x such that straight f open parentheses x close parentheses equals straight f to the power of negative 1 space end exponent open parentheses x close parentheses.

This will be the point at which the two graphs meet.

The point will be on the line y equals x so there is no need to work out straight f to the power of negative 1 end exponent open parentheses x close parentheses.

By sketching the graph in part (a) this point (with coordinates open parentheses 2 comma space 2 close parentheses) should have already been considered. Only the x value is required.

bold italic x bold equals bold 2

The x value could also be found by solving straight f open parentheses x close parentheses equals x.

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    Paul

    Author: Paul

    Expertise: Maths Content Creator

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.