“Teach A Level Maths” Vol. 2: A2 Core Modules 3: Graphs of Inverse Functions © Christine Crisp

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Consider the graph of the function The inverse function is

Consider the graph of the function x x x x The inverse function is An inverse function is just a rearrangement with x and y swapped. So the graphs just swap x and y!

What else do you notice about the graphs? x x is a reflection of in the line y = x The function and its inverse must meet on y = x

e.g. On the same axes, sketch the graph of and its inverse. N.B! Solution: x

e.g. On the same axes, sketch the graph of and its inverse. N.B! Solution: N.B. Using the translation of we can see the inverse function is .

A bit more on domain and range The previous example used . The domain of is . Since is found by swapping x and y, the values of the domain of give the values of the range of . Domain Range

A bit more on domain and range The previous example used . The domain of is . Since is found by swapping x and y, the values of the domain of give the values of the range of . Similarly, the values of the range of give the values of the domain of

SUMMARY The graph of is the reflection of in the line y = x. It follows that the curves meet on y = x At every point, the x and y coordinates of become the y and x coordinates of . The values of the domain and range of swap to become the values of the range and domain of . e.g.

A Rule for Finding an Inverse e.g. 1 An earlier example sketched the inverse of the function There was a reason for giving the domain as . Let’s look at the graph of for all real values of x.

x = 1, y = 1 . . . x = 3, y = 1 This function is many-to-one. e.g. and An inverse function undoes a function. But we can’t undo y = 1 since x could be 1 or 3.

x = 1, y = 1 . . . x = 3, y = 1 This function is many-to-one. e.g. and An inverse function undoes a function. But we can’t undo y = 1 since x could be 1 or 3.

x = 1, y = 1 . . . x = 3, y = 1 This function is many-to-one. e.g. and An inverse function undoes a function. An inverse function only exists if the original function is one-to-one.

We can have either If a function is many-to-one, the domain must be restricted to make it one-to-one.

or If a function is many-to-one, the domain must be restricted to make it one-to-one.

e.g. 2 Find possible values of x for which the inverse function of can be defined. Solution: Let’s sketch the graph of for The function is clearly many-to-one so we must find a domain that gives us a section that is one-to-one. The most obvious section to use is the part close to the origin.

e.g. 2 Find possible values of x for which the inverse function of can be defined. Solution: Let’s sketch the graph of for Solution: The function is clearly many-to-one so we must find a domain that gives us a section that is one-to-one. The most obvious section to use is the part close to the origin.

e.g. 2 Find possible values of x for which the inverse function of can be defined. Solution: Let’s sketch the graph of for Solution: The function is clearly many-to-one so we must find a domain that gives us a section that is one-to-one. The most obvious section to use is the part close to the origin.

e.g. 2 Find possible values of x for which the inverse function of can be defined. Solution: Let’s sketch the graph of for The function is clearly many-to-one so we must find a domain that gives us a section that is one-to-one. These values are called the principal values. In degrees, the P.Vs. are

Exercise Suggest principal values for and ( Give your answers in both degrees and radians ) Solution:

Exercise Suggest principal values for and ( Give your answers in both degrees and radians ) Solution: or

or

SUMMARY Only one-to-one functions have an inverse function. If a function is many-to-one, the domain must be restricted to make the function one-to-one. The restricted domains of the trig functions are called the principal values. radians degrees

Exercise 1 (a) Sketch the function where . (b) Write down the range of . (c) Suggest a suitable domain for so that the inverse function can be found. (d) Find and write down its domain and range. (e) On the same axes sketch .

Solution: (a) (b) Range of : (c) Restricted domain: ( We’ll look at the other possibility in a minute. ) (d) Inverse: Let Rearrange: Swap: Domain: Range:

Solution: (b) Range of : (a) (c) Suppose you chose for the domain (d) Let As before Rearrange: We now need since

Solution: (a) Range: (b) (c) Suppose you chose for the domain Choosing is easier! (d) Let As before Rearrange: We now need since Swap: Domain: Range:

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

SUMMARY At every point, the x and y coordinates of become the y and x coordinates of . The values of the domain and range of swap to become the values of the range and domain of . e.g. The graph of is the reflection of in the line y = x. It follows that the curves meet on y = x

or For we can have: An inverse function undoes a function. An inverse function only exists if the original function is one-to-one. If a function is many-to-one, the domain must be restricted to make it one-to-one. either

e.g. 1 Find possible values of x for which the inverse function of can be defined. The function is clearly many-to-one so we must find a domain that gives us a section that is one-to-one. Let’s sketch the graph of for Solution:

These values are called the principal values. In degrees, the P.Vs. are The part closest to the origin is used for the domain.

SUMMARY Only one-to-one functions have an inverse function. If a function is many-to-one, the domain must be restricted to make the function one-to-one. The restricted domains of the trig functions are called the principal values. radians degrees