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

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

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−1 𝑔 𝑥 = 𝑥 2 2𝑥−1 + 𝑥 2 𝑓+𝑔 𝑥 = 𝑥 2 +2𝑥−1

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−1 𝑔 𝑥 = 𝑥 2 2𝑥−1 − 𝑥 2 𝑓−𝑔 𝑥 = −𝑥 2 +2𝑥−1

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−1 𝑔 𝑥 = 𝑥 2 2𝑥−1 ∗ 𝑥 2 𝑓∗𝑔 𝑥 =2 𝑥 3 − 𝑥 2

*(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−1 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 )

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

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.

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.

Looking at Inverses Graphically

Looking at Inverses Graphically

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

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