1. Inverse Functions

2. Inverse Functions Functions
One-to-one Function
Functions
A function is a mapping whereby every element in the domain is mapped to only 1 element in the range.
ie) Whatever number you start with, there is only 1 possible answer to the operation performed on it.
An example of a mapping which is not a function would be square rooting, where the starting number may result in no answer, or 2 answers.
Set A
Set B
eg) f(x) = x + 5
eg) f(x) = 3x - 2
Many-to-one Function
Set A
Set B
eg) f(x) = x2 + 1
eg) f(x) = 6 - 3x2
Not a function
Set A
Set B
eg) f(x) = √x
eg) f(x) = 1/x 2B

3. ‘A value in the domain (x) gets mapped to one value in the range’
Functions
One-to-one Function
Functions
A function is a mapping whereby every element in the domain is mapped to only 1 element in the range.
ie) Whatever number you start with, there is only 1 possible answer to the operation performed on it.
An example of a mapping which is not a function would be square rooting, where the starting number may result in no answer, or 2 answers.
‘A value in the domain (x) gets mapped to one value in the range’
Many-to-one Function
‘Multiple values in the domain (x) get mapped to the same value in the range’
Not a function
‘A value in the range can be mapped to none, one or more values in the range’ 2B

4. Note:
For a function to have an inverse, the members of its domain and range must be of one-to-one correspondence

5. Functions 2E 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
Inverse
f(x) = x + 4
f-1(x) = x - 4
g(x) = 2x
g-1(x) = x/2
h(x) = 4x + 2
h-1(x) = x – 2/4 2E

6. Functions 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
f(x) = 3x2 - 4
y = 3x2 - 4
+ 4
y + 4 = 3x2
÷ 3
(y + 4)/3 = x2
Square root
√((y + 4)/3) = x
The inverse is written ‘in terms of x’
f-1(x) = √((x + 4)/3) 2E

7. Functions 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) = 3/(x – 1)
y = 3/(x – 1)
Multiply by (x – 1)
y(x – 1) = 3
Multiply the bracket
yx - y = 3
Add y
yx = 3 + y
Divide by y
x = (3 + y)/y
The inverse is written ‘in terms of x’
m-1(x) = (3 + x)/x 2E

8. The inverse is written ‘in terms of x’
Functions
Finding f-1(x)
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)
‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’
‘Find f-1(x), stating its domain. Sketch the graphs and describe the link between them.
f(x) = √(x – 2)
y = √(x – 2)
Square
y2 = x - 2
Add 2
y = x
f-1(x) = x2 + 2
The inverse is written ‘in terms of x’ 2E

9. Finding the domain of f-1(x)
Functions
Finding the domain of f-1(x)
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)
‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’
‘Find f-1(x), stating its domain. Sketch the graphs and describe the link between them.
f-1(x) = x2 + 2
‘The domain and range of a function switch around for its inverse’
f(x)
f(x) = √(x – 2) x
Range for f(x) 
f(x) ≥ 0 or y ≥ 0
Domain for f-1(x) 
x ≥ 0

10. Sketching the graph of f-1(x)
Functions
Sketching the graph of f-1(x)
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)
‘The function f(x) is defined by f(x) = √(x – 2), x ε R, x ≥ 2’
‘Find f-1(x), stating its domain. Sketch the graphs and describe the link between them.
f-1(x) = x2 + 2, {x ε R, x ≥ 0}
Domain is x ≥ 0, so we can draw the graph for any values of x in this range
f-1(x) = x2 + 2
f(x)
f(x) = √(x – 2) x
The link is that f(x) is reflected in the line y = x 2E

11. ‘The domain and range of a function switch around for its inverse’
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)
If g(x) is defined as:
g(x) = 2x – 4, {x ε R, x ≥ 0},
Calculate and sketch g(x) and g-1(x), stating the domain of g-1(x).
g(x) = 2x - 4
g-1(x) = (x + 4)/2
Domain  x ≥ 0
Domain  x ≥ -4
Range  g(x) ≥ -4
Range  g-1(x) ≥ 0
g(x)
f(x)
g-1(x) x
The link is that f(x) is reflected in the line y = x

12. Pg 25 Exercise 4
Questions 1, 2, 8, 9
For homework look at slide 9,10,11 and make a final note on inverse functions in your jotter.