3: Graphs of Inverse Functions Christine Crisp Teach A Level
Maths Vol. 2: A2 Core Modules Slide 2 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" Slide 3 Inverse Functions Consider
the graph of the function The inverse function is Slide 4 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 Slide 5 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 Slide 6 Inverse Functions e.g.On the same axes,
sketch the graph of and its inverse. N.B! x Solution: Slide 7
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. Slide 8 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. Slide 9 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 Slide 10 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. Slide 11 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. Slide 12 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. Slide 13 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. Slide 14 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. Slide 15 Inverse Functions If a function is
many-to-one, the domain must be restricted to make it one-to-one.
We can have either Slide 16 Inverse Functions If a function is
many-to-one, the domain must be restricted to make it one-to-one.
or Slide 17 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. Slide 18
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. Slide 19 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. Slide 20 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. Slide 21 Inverse Functions (
Give your answers in both degrees and radians ) Exercise Suggest
principal values for and Solution: Slide 22 Inverse Functions (
Give your answers in both degrees and radians ) Exercise Suggest
principal values for and Solution: or Slide 23 Inverse Functions
Slide 24 or Slide 25 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 Slide 26 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.
Slide 27 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 : Slide 28
Inverse Functions Solution: (a) Rearrange: (d) Let As before (c)
Suppose you chose for the domain We now need since (b) Range of :
Slide 29 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! Slide 30 Inverse
Functions Slide 31 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. Slide 32 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 Slide 33 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 Slide 34 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: Slide 35 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. Slide 36 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
