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

2. 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.

3. 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

4. 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 =

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

6. 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.

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

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

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

10. Inverse Functions
Inverse Function:

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

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

13. 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).

14. 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).

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

16. 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.