2. Objectives of the Lesson
• Explain the concept of an ‘inverse function’ by using a mapping
diagram
• Demonstrate how the inverse of a function is found.
3. The Concept
The function f (x) = 2x +5 maps the domain elements to the range
elements in the diagram shown.
i.e. 𝑓 1 = 2 1 + 5 = 7
You substitute the domain (input) elements into the function in order
to obtain the range (output) elements.
If, however, we were to map the range elements to the domain
elements then we would need another function that reverses the first
function. That function is known as the inverse function.
4. 𝑓−1
𝑥 =
𝑥 − 5
2
e.g. 𝑓−1 7 =
7 −5
2
=
2
2
= 1
In the inverse function, the range elements become the
domain elements and vice versa.
Such a fact is very important when determining the inverse
function of a given function
6. Solutions
The rules used for transposition will become very useful for this topic.
a) f(x) = 3x – 5
First: rewrite the function as y = 3x – 5
Then: interchange x for y and y for x (x y), i.e. x = 3y - 5
Finally: transpose for y (make y the subject)
y was multiplied by 3, then 5 was subtracted
Reversing the process we get: add 5 but show it on the other side of function
x + 5 = 3y
Now divide by 3 (show it on the opposite side):
𝑥+5
3
= 𝑦
Rewrite the inverse function using the notation: 𝑓−1 𝑥 =
𝑥+5
3