The document discusses deriving the inverse of a composite function. It provides examples of composite functions and their inverses. Specifically: - It defines a composite function as first applying one function, then another, denoted fg. The inverse of a composite function is found by first deriving the composite function, then taking its inverse. - An example problem finds the inverses of (fg), (gf), and (hg), where f, g, and h are functions of x. The composite functions are derived then their inverses are obtained by swapping x and y. - The inverse of (fg) is (3x+16)/3. The inverse of (gf) is (3x+

1. Deriving the Inverse of a
Composite Function
Presented by:
ALONA HALL

2. Objectives of the Lesson
• Identify the steps involved in deriving the inverse of a composite
function
• Demonstrate how the inverse of a composite 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

5. Inverse of Composite Functions
When finding the inverse of a composite function, we first derive the composite
function then find its inverse. For example, the notation (𝑓𝑔)−1 means, derive the
composite function fg then find its inverse.
𝑁𝑒𝑣𝑒𝑟 𝑐𝑜𝑛𝑓𝑢𝑠𝑒 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑎 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒
composition of inverse functions denoted 𝑓−1 𝑔−1.
𝑇ℎ𝑒 𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡.
A clear understanding of the concepts related to inverse and composition will
aid any student to distinguish between the two. The student would also be able
to manipulate any given variation whether related to composition, inverse
𝑜𝑟 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.

6. Example
𝐺𝑖𝑣𝑒𝑛 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠, 𝑓 𝑥 =
3𝑥 + 5
3
, g 𝑥 = 2𝑥 − 7 𝑎𝑛𝑑 ℎ 𝑥 =
13𝑥
𝑥 + 2
𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔:
a) (𝑓𝑔)−1
b) (𝑔𝑓)−1
c) (ℎ𝑔)−1

7. Solutions
a) (𝑓𝑔)−1
Recall: 𝑓 𝑥 =
3𝑥+5
3
, g 𝑥 = 2𝑥 − 7 𝑎𝑛𝑑 ℎ 𝑥 = 13𝑥
• First find 𝑓𝑔(𝑥), then derive its inverse.
• 𝑓𝑔 𝑥 = 𝑓 𝑔 𝑥
=
3 𝑔 𝑥 + 5
3
=
3 2𝑥 − 7 + 5
3
=
6𝑥 − 21 + 5
3
𝑓𝑔(𝑥) =
6𝑥 − 16
3
𝑁𝑜𝑤 𝑓𝑖𝑛𝑑 (𝑓𝑔)−1.
• 𝑦 =
6𝑥 −16
3
• 𝑥 ↔ 𝑦: 𝑥 =
6𝑦 −16
3
• Transpose for y
3𝑥 = 6𝑦 − 16
3𝑥 + 16 = 3𝑦
3𝑥 + 16
3
= 𝑦
∴ (𝑓𝑔 )−1
=
3𝑥 + 16
3

8. Solutions
𝑁𝑜𝑤 𝑓𝑖𝑛𝑑 (𝑔𝑓) −1
𝑦 =
6𝑥 − 11
3
𝑥 ↔ 𝑦: 𝑥 =
6𝑦 − 11
3
𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑠𝑒 𝑓𝑜𝑟 𝑦
3𝑥 = 6𝑦 − 11
3𝑥 + 11 = 6𝑦
3𝑥 + 11
6
= 𝑦
(𝑔𝑓)−1=
3𝑥 + 11
6
b) (𝑔𝑓)−1
𝐹𝑖𝑟𝑠𝑡 𝑓𝑖𝑛𝑑 𝑔𝑓 𝑥 = 𝑔 𝑓 𝑥
= 2 𝑓 𝑥 − 7
=
2
1
3𝑥 + 5
3
− 7
=
6𝑥 + 10
3
−
7
1
=
1 6𝑥 + 10 − 3(7)
3
=
6𝑥 + 10 − 21
3
𝑔𝑓 𝑥 =
6𝑥 − 11
3

9. Solution
c) (ℎ𝑔)−1
𝑓𝑖𝑟𝑠𝑡 𝑓𝑖𝑛𝑑 ℎ𝑔 𝑥 =
ℎ[𝑔(𝑥)] =
13 [𝑔(𝑥)]
𝑔 𝑥 + 2
=
13(2𝑥 − 7)
2𝑥 − 7 + 2
ℎ[𝑔(𝑥)] =
26𝑥 − 91
2𝑥 − 5
𝑁𝑜𝑤 𝑓𝑖𝑛𝑑(ℎ𝑔)−1
𝑦 =
26𝑥 − 91
2𝑥 − 5
𝑥 ↔ y: x =
26𝑦 − 91
2𝑦 − 5
transpose for y
𝑥 2𝑦 − 5 = 26𝑦 − 91
2𝑥𝑦 − 5𝑥 = 26𝑦 − 91
Group the y terms on one side and the non-y terms on the other
2𝑥𝑦 − 26𝑦 = −91 + 5𝑥
Factor out y: 𝑦 2𝑥 − 26 = 5𝑥 − 91
𝑦 =
5𝑥 − 91
2𝑥 − 26
ℎ𝑔−1
=
5𝑥 − 91
2𝑥 − 26