Inverse Functions (Cambridge International IGCSE Further Maths)

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 $f^{-1}(x)=\dots$ or f ^ - 1 colon x ->from bar …

  • For example, if $f(x)=2x+1$ its inverse can be written as
  • $f^{-1}(x)=\frac{(x--1)}{2}$ or $\mathrm{f}^{- 1} : x \rightarrow \text{from} \text{bar} \frac{( x - 1 )}{2}$

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 form$y=\dots$ e.g. $y=2x+1$

• STEP 2 Swap the $x$'s and$y$'s to get $x=\dots$ e.g. $x=2y+1$

• STEP 3 Rearrange the expression to make $y$ the subject again $\begin{gathered} x - 1 & = 2 y \\ \frac{x - 1}{2} & = y \rightarrow y = \frac{x - 1}{2} \end{gathered}$

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

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

  • $y$should not exist in the final answer

    • e.g. $\mathrm{f}^{- 1} ( x ) = \frac{ x - 1}{2}$

How does a function relate to its inverse?

• If $\mathrm{f} ( 3 ) = 10$ 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

    • If $\mathrm{f} ( 3 ) = 10$ then $\mathrm{f}^{- 1} ( 10 ) = 3$

• $ff^{- 1} ( x ) = \mathrm{f}^{- 1} \mathrm{f} ( x ) = x$

  • 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 $\mathrm{f} ( x ) = 2^{x}$ and you want to solve $\mathrm{f}^{- 1} ( x ) = 5$

  • Finding the inverse function $\mathrm{f}^{- 1} ( x )$ in this case is tricky (impossible if you haven't studied logarithms)
  • instead, take f of both sides and use that $\mathrm{ff}^{-1}$ cancel each other out:

$\begin{gathered} ff^{- 1} ( x ) & = \mathrm{f} ( 5 ) \\ x = \mathrm{f} ( 5 ) \\ x = 2^{5} = 32 \end{gathered}$

What condition is needed for an inverse function to exist?

• For the inverse function to exist, $\mathrm{f}^{- 1} ( x )$, the original function$\mathrm{f} ( x )$ must be one-to-one

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

  • Substituting this 1 output into $f^{-1}$ 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 $\mathrm{f} ( x ) = 5 - 3 x$.

Write the function in the form $y=5-3x$ and then swap the $x$ and $y.$* *

$y=5-3x\,x=5-3y$

*Rearrange **the expression to make *$y$ the subject again.

$\begin{gathered} x & = 5 - 3 y \\ x + 3 y & = 5 \\ 3 y & = 5 - x \\ y & = \frac{5 - x}{3} \end{gathered}$

Rewrite using the correct notation for an inverse function.

$f^{-1}(x)=\frac{5-x}{3}$

Domain & Range of Inverse Functions

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

• 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 $f(x)=\sqrt{3x-2},x>\frac{3}{2}$.

Write down the domain and range of $f^{-1}(x)$.

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

The range of $\mathrm{f} ( x )$ is

$f>0$

Final answer: $\therefore$T he domain of $f^{-1}(x)$is$x>0$

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

Final answer: $\therefore$The range of$f^{-1}(x)$is$f^{-1}>\frac{3}{2}$

Graphs of Inverse Functions

How are the graphs of a function and its inverse related?

• The graph of an inverse function, $y = \mathrm{f}^{- 1} ( x )$, is a reflection of the graph of the function, $y = \mathrm{f} ( x )$, in the line $y=x$

• Key features of the graph of $y = \mathrm{f} ( x )$ 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 $y=x$, and if need be, the graph of $y = \mathrm{f} ( x )$

• STEP 2

  • Reflect the graph of $y = \mathrm{f} ( x )$ in the line $y=x$

    • 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 $( 4 , 0 )$ will be reflected to the point $( 0 , 4 )$

  • Consider any asymptotes on the graph of $y = \mathrm{f} ( x )$ - these will also be need reflecting

    • e.g. The asymptote (line) $x=-2$ will be reflected to the line $y=-2$

  • Consider any restrictions on the domain and range of $\mathrm{f} ( x )$

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

• STEP 3

  • Label key points on the sketch of $y = \mathrm{f}^{- 1} ( x )$ and state the equations of any asymptotes

• This process works the other way round - the graph of $y = \mathrm{f} ( x )$ can be sketched from the graph of $y = \mathrm{f}^{- 1} ( x )$

Exam Hint

• If not given, sketch the graphs of $y = \mathrm{f} ( x )$ and $y=x$ to help sketch the graph of the inverse, $y = \mathrm{f}^{- 1} ( x )$

• If the graph of $y = \mathrm{f} ( x )$ is given you do not need to know the expression for $\mathrm{f} ( x )$ to sketch $y = \mathrm{f}^{- 1} ( x )$

  • Just reflect whatever is given in the line $y=x$

Worked Example The diagram below shows the graph of $y = \mathrm{f} ( x )$, where $\mathrm{f} ( x ) = 4 - \frac{4}{x} , x > 0$.

a)On a copy of the diagram, sketch the graph of $y = \mathrm{f}^{- 1} ( x )$. 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 $y=x$.

Draw the line $y=x$ to help sketch the inverse function.

The $x$-axis intercept $( 1 , 0 )$ becomes the $y$-axis intercept, $( 0 , 1 )$.

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

b) Use your sketch, or otherwise, to write down the value of $x$ such that $\mathrm{f} ( x ) = \mathrm{f}^{- 1 } ( x )$.

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

The point will be on the line $y=x$ so there is no need to work out $\mathrm{f}^{- 1} ( x )$.

By sketching the graph in part (a) this point (with coordinates $( 2 , 2 )$) should have already been considered. Only the $x$ value is required.

$x=2$

The$x$value could also be found by solving$\mathrm{f} ( x ) = x$.