1. Sum, Difference, Product, Quotient, Composition and Inverse
Combining Functions
Sum, Difference, Product, Quotient, Composition and Inverse

2. Warm Up: 10/7 𝑓 𝑥 =3 𝑥 2 +𝑥−5 g 𝑥 = −2𝑥+6 −4 Find 𝑓(−2).
𝑓 𝑥 =3 − −2 −5=5
g 𝑥 = −2𝑥+6 −4
Find g(1).
g 𝑥 = − −1=2−4=−2
When the transformation is on the inside it affects the...
When the transformation is on the outside it affects the…

3. Sum 𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔(𝑥) Example: 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 = 𝑥 2 2𝑥−1 + 𝑥 2
𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔(𝑥)
Example:
𝑓 𝑥 =2𝑥− 𝑔 𝑥 = 𝑥 2
2𝑥−1 + 𝑥 2
𝑓+𝑔 𝑥 = 𝑥 2 +2𝑥−1

4. Difference 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔(𝑥) Example: 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 = 𝑥 2 2𝑥−1 − 𝑥 2
𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔(𝑥)
Example:
𝑓 𝑥 =2𝑥− 𝑔 𝑥 = 𝑥 2
2𝑥−1 − 𝑥 2
𝑓−𝑔 𝑥 = −𝑥 2 +2𝑥−1

5. Product 𝑓∗𝑔 𝑥 =𝑓 𝑥 ∗𝑔(𝑥) Example: 𝑓 𝑥 =2𝑥−1 𝑔 𝑥 = 𝑥 2 2𝑥−1 ∗ 𝑥 2
𝑓∗𝑔 𝑥 =𝑓 𝑥 ∗𝑔(𝑥)
Example:
𝑓 𝑥 =2𝑥− 𝑔 𝑥 = 𝑥 2
2𝑥−1 ∗ 𝑥 2
𝑓∗𝑔 𝑥 =2 𝑥 3 − 𝑥 2

6. *(you can leave it as: 𝟐𝒙−𝟏 𝒙 𝟐 )
Quotient
𝑓 𝑔 𝑥 = 𝑓 𝑥 𝑔 𝑥 , provided that 𝑔(𝑥)≠0.
Example:
𝑓 𝑥 =2𝑥− 𝑔 𝑥 = 𝑥 2
2𝑥−1 𝑥 2 = 2𝑥 𝑥 2 − 1 𝑥 2
𝑓 𝑔 𝑥 = 2 𝑥 − 1 𝑥 2
*(you can leave it as: 𝟐𝒙−𝟏 𝒙 𝟐 )

7. Composition of Functions
𝑓°𝑔 𝑥 =𝑓(𝑔 𝑥 )
Example:
𝑓 𝑥 =2𝑥− 𝑔 𝑥 = 𝑥 2
𝑓°𝑔 𝑥 =2 𝑥 2 −1
𝑔°𝑓 𝑥 = (2𝑥−1) 2 =4 𝑥 2 −4𝑥+1

8. Inverse Function 𝑓 −1 𝑏 =𝑎 if and only if 𝑓 𝑎 =𝑏
If 𝑓 is a one-to-one function with domain 𝐷 and range 𝑅, then the inverse function of 𝒇, denoted 𝑓 −1 , is the function with domain 𝑅 and range 𝐷 defined by:
𝑓 −1 𝑏 =𝑎 if and only if 𝑓 𝑎 =𝑏
One-to-one: A function in which each element of the range corresponds to exactly one element in the domain.

9. Inverses Algebraically Graphically
Find an equation for 𝑓 −1 𝑥 if 𝑓 𝑥 = 𝑥 𝑥+1 .
Step 1: Switch 𝑥 and 𝑦.
𝑥= 𝑦 𝑦+1
Step 2: Solve for 𝑦.
𝑥 𝑦+1 =𝑦
𝑥𝑦+𝑥=𝑦
𝑥=𝑦−𝑥𝑦
𝑥=𝑦 1−𝑥
𝑦= 𝑥 1−𝑥 = 𝑓 −1 𝑥
The points (𝑎,𝑏) and (𝑏,𝑎) in the coordinate plane are symmetric with respect to the line 𝒚=𝒙. The points (𝑎,𝑏) and (𝑏,𝑎) are reflections of each other in the line 𝒚=𝒙.

10. Looking at Inverses Graphically

11. Looking at Inverses Graphically

12. Inverses: reflected over the line 𝒚=𝒙.
𝑦= 𝑒 𝑥 and y=ln⁡(𝑥)

13. Thus, 𝑓(𝑥) and 𝑔(𝑥) are inverses.
𝒇 𝒙 = 𝒙 𝟑 +𝟏
𝒈 𝒙 = 𝟑 𝒙−𝟏
𝑓 𝑔 𝑥 = ? = 3 𝑥− =𝑥−1+1 =𝑥
g 𝑓 𝑥 = ? = 3 ( 𝑥 3 +1)−1 = 3 𝑥 3 =𝑥
Thus, 𝑓(𝑥) and 𝑔(𝑥) are inverses.