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

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Module C3 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

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Inverse Functions Consider the graph of the function The inverse function is

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

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

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

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Inverse Functions 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.

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

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

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Inverse Functions 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.

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Inverse Functions 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. Lets look at the graph of for all real values of x.

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Inverse Functions This function is many-to-one. e.g. x = 1, y = 1... and x = 3, y = 1 An inverse function undoes a function. But we cant undo y = 1 since x could be 1 or 3.

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Inverse Functions This function is many-to-one. e.g. x = 1, y = 1... and x = 3, y = 1 An inverse function undoes a function. But we cant undo y = 1 since x could be 1 or 3.

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

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

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

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

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Inverse Functions 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. Solution: Lets sketch the graph of for e.g. 2 Find possible values of x for which the inverse function of can be defined.

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Inverse Functions 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. Solution: Lets sketch the graph of for e.g. 2 Find possible values of x for which the inverse function of can be defined.

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Inverse Functions 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 Solution:Lets sketch the graph of for e.g. 2 Find possible values of x for which the inverse function of can be defined.

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

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

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Inverse Functions

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or

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Inverse Functions 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

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

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

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

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

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Inverse Functions

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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.

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Inverse Functions 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

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Inverse Functions 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

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Inverse Functions 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. Lets sketch the graph of for Solution:

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

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Inverse Functions 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

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