Inverse functions
Domain of f (x)
Range of f (x)
f ( x)  2 x  1
1
3
2
5
3
7
4
9
5
11
Range of f
1
( x)
f 1 ( x)
x 1
2
Domain of
f
1
( x)
For a unique inverse to exist the function
must be one-one for the given domain.
Many-one functions can have an inverse
when the domain is restricted, so that part
of the domain will be one-one.
Inverse Functions
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse
is f-1(x)
Some simple inverses
Function
f(x) = x + 4
g(x) = 2x
h(x) = 4x + 2
Inverse
f-1(x) = x - 4
g-1(x) = x/2
h-1(x) = x – 2/4
2E
Inverse Functions
Find the inverse of the
following function
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse
is f-1(x)
To calculate the inverse of a
function, you need to make ‘x’ the
subject
f(x) = 3x2 - 4
y = 3x2 - 4
y + 4 = 3x2
y + 4/
3
= x2
√(y + 4/3) = x
The inverse is written
‘in terms of x’
+4
÷3
Square
root
f-1(x) = √(x + 4/3)
2E
Find the inverse of the
following function
Inverse Functions
You need to be able to work out the
inverse of a given function.
If f(x) is the function, the inverse
is f-1(x)
To calculate the inverse of a
function, you need to make ‘x’ the
subject
m(x) =
y =
3/
(x – 1)
y(x – 1) = 3
yx - y = 3
yx = 3 + y
The inverse is written
‘in terms of x’
x=
3 + y/
y
3/
(x – 1)
Multiply by
(x – 1)
Multiply the
bracket
Add y
Divide
by y
m-1(x) = 3 + x/x
2E
Graphs of inverse functions
1.
f ( x)  2 x  1
Reflections of each other in
2.
f ( x)  x  1
3.
f ( x)  3 x  4
4.
f ( x)  x
2
yx
Given f ( x)  3x  4 x  R find the
1
inverse function f ( x), x  R
1
f ( x) 
2 x
Given
x  R, x  2 find
the inverse function
From the graph
Domain
Range
1
f ( x) 
2 x
x  R, x  2
y  R, y  0
2x 1
f ( x) 
x
x  R, x  0
y  R, y  2
1
Given
13  6 x
f ( x) 
7
xR
A) Find the inverse function
 7 x  13
f ( x) 
6
1
Given
13  6 x
f ( x) 
7
xR
B) Verify the result using x = 2
Given
13  6 x
f ( x) 
7
C) Show that
1
xR
ff ( x)  f
1
f ( x)  x
Self-inverse functions
1
f ( x)  f ( x)
Show that
is self-inversed
x
f ( x) 
x  R, x  1
x 1
Homework:
Exercise 2C page 29
Numbers:
2, 4, 5, 6, 12, 13
Even and Odd functions
f ( x)  f (  x)
The graph has 180
degrees rotational
symmetry about
the origin (0,0)
f ( x)  x 2
f ( x)   f (  x)
f ( x)  x 4
f ( x)  x
f ( x)  cos x
f ( x)  x 3
The graph is
symmetrical about
the y axis
f ( x)  sin x