MCR3U1
U2L5
THE INVERSE FUNCTION
(and its Properties)
PART A ~ INTRODUCTION
The inverse function is the reverse of the original function.
It “undoes” what the original function has done and can be
found using the inverse operations of the original function
in reverse order.
What is the inverse operation for addition?
_____________________________
What is the inverse operation for multiplication?
_____________________________
What is the inverse operation for squaring a number?
_____________________________
ex.
Consider the function, 𝑦 = 5𝑥 + 8.
If the order of operations in the function are “multiply by 5 and then add 8”
then the inverse order of operations are “subtract 8 and then divide by 5”.
PART B ~ THE INVERSE FUNCTION DEFINED
 If 𝑦 = 𝑓(𝑥), the inverse is denoted ________________________________.
(This notation can only be used when the inverse is a function.)
 The domain of a function is equal to the range of its inverse and vice versa.
 The graph of the inverse is a reflection of the original function in the line, 𝑦 = 𝑥.
 The inverse of a function is NOT necessarily a function itself.
PART C ~ DETERMINING THE INVERSE USING MAPPING
If (𝑥, 𝑦) is any point on 𝑦 = 𝑓(𝑥), then (𝑦, 𝑥) is any point on 𝑦 = 𝑓 −1 (𝑥).
In other words, points can be translated using the mapping:
(𝑥, 𝑦) → (𝑦, 𝑥)
Ex.
(2,5) 
Example 
Ex.
If f(1) = 6, then 𝑓 −1 (6) =
Determine the inverse of f = { (1, –3), (–2, 5), (4, 0) }.
MCR3U1
U2L5
PART D ~ DETERMINING THE INVERSE GRAPHICALLY
Reflection Property: The graph of the inverse is a reflection of the original
function in the line, 𝑦 = 𝑥.
Example 
Use the given graph to obtain the graph of the inverse:
a)
b)
Which of the above inverses is still a function???
PART E ~ DETERMINING THE INVERSE ALGEBRAICALLY
Methods of Determining the Inverse:
i)
Interchange method:
ii)
"Short-cut" method (tally chart).
Example 
a)
1.
2.
3.
4.
Replace f(x) with y.
Interchange x and y in the relation.
Solve for y.
Replace y with 𝑓 −1 (𝑥) (if allowed).
Determine the inverse of each function:
𝑓(𝑥) = 4𝑥 − 3, then evaluate 𝑓 −1 (5);
𝑓(𝑥)
𝑓 −1 (𝑥)
MCR3U1
b)
U2L5
𝑔(𝑥) =
12−2𝑥
3
𝑔(𝑥)
𝑔−1 (𝑥)
Example 
Archaeologists use models for the relationship between height and footprint
length to determine the height of a person based on the lengths of the bones
they discover. The relationship between height, ℎ(𝑥), in centimeters and
footprint length, x, in centimeters is given by ℎ(𝑥) = 1.1𝑥 + 143.6.
a)
Determine the inverse equation to express the length of a footprint
in terms of the height.
b)
If a person is 170 cm tall, determine the length of their footprint.
HOMEWORK: p.46–49 #1, 2abe, 3, 4d, 5bdf, 6bcdf, 8 (don’t sketch), 9abcde, 10de, 12, 16, 17, 20