Operations on Functions, Compositions, and Inverses Lesson 6.6 Operations on Functions, Compositions, and Inverses

Lesson Objectives At the end of the lesson, students can: Perform operations (addition, subtraction, multiplication, division, composition) on functions. Find the correct domain for composite functions. Recognize and state implicitly defined functions. Define and find inverse relations and inverse functions. Find the correct domain for inverse relations and inverse functions.

Operations on Functions If f and g are 2 functions, then 𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔(𝑥) 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔(𝑥) 𝑓𝑔 𝑥 =𝑓 𝑥 𝑔(𝑥) 𝑓 𝑔 𝑥 = 𝑓(𝑥) 𝑔(𝑥) , 𝑔(𝑥)≠0

Operations on Functions 𝑓 𝑥 = 𝑥−1 2 and 𝑔 𝑥 =3−𝑥 , find and state domain of: 𝑓+𝑔 𝑥 = 𝑓−𝑔 𝑥 = 𝑓𝑔 𝑥 = 𝑓 𝑔 𝑥 =

Composition of Functions 𝑓 𝑥 = 𝑥 2 −1 𝑔 𝑥 = 𝑥 Find (𝑓°𝑔)(2).

Composition of Functions 𝑓°𝑔 𝑥 =𝑓 𝑔(𝑥) Domain of a composition: The domain of the “inside function” and the domain of the answer (final) are concerns. 𝑓 𝑥 = 𝑥 2 −1 𝑔 𝑥 = 𝑥 Find (𝑓°𝑔)(𝑥) and state the domain.

Composition of Functions 𝑓 𝑥 = 𝑥 2 −1 𝑔 𝑥 = 𝑥 Find (𝑔°𝑓)(𝑥) and state the domain.

Composition of Functions 𝑓 𝑥 = 1 𝑥 2 −4 𝑔 𝑥 =𝑥+ 1 𝑥 Find (𝑓°𝑔)(𝑥) and state the domain.

One-to-One Functions 1 – to – 1 Function:

Inverse Functions Inverse Function:

Inverse Functions If 𝑓 𝑥 = 𝑥 𝑥+1 , find an equation for 𝑓 −1 (𝑥)

Verifying Algebraic Inverses Show 𝑓 𝑔 𝑥 =𝑔 𝑓 𝑥 =𝑥 𝑓 𝑥 = 𝑥 3 +1 𝑔 𝑥 = 3 𝑥−1

Inverse Functions Show that 𝑓 𝑥 = 𝑥+3 has an inverse function. Find a rule for 𝑓 −1 𝑥 . State any restrictions inherited from 𝑓(𝑥).

Inverse Functions Show that 𝑓 𝑥 = 𝑥+3 𝑥−2 has an inverse function. Find a rule for 𝑓 −1 𝑥 . State any restrictions inherited from 𝑓(𝑥).

Homework Read pp. 113 - 127 Do p. 127: 1 – 19 odd, 27 – 61 odd

Lesson Objectives At the end of the lesson, students can: Perform operations (addition, subtraction, multiplication, division, composition) on functions. Find the correct domain for composite functions. Recognize and state implicitly defined functions. Define and find inverse relations and inverse functions. Find the correct domain for inverse relations and inverse functions.