1. Inverse Functions
𝑓 −1 "𝑓 𝑖𝑛𝑣𝑒𝑟𝑠𝑒"

2. 5 Important Facts 𝑦= 𝑒 𝑥 𝑦=ln x
The domain and range of 𝐟 and 𝐟 −𝟏 are swapped.
This means that if −𝟐, 𝟒 is on 𝒇, (𝟒, −𝟐) must be on 𝒇 −𝟏 .
𝑦= 𝑒 𝑥
Domain: −∞,∞
Range: (0,∞)
𝑦=ln x
Domain: 0,∞
Range: (−∞,∞)

3. 5 Important Facts
2. A function 𝒇 has an inverse if and only if it passes the Horizontal Line Test.

4. 5 Important Facts 𝒚= 𝒙 𝟑 & 𝐲= 𝟑 𝒙 𝒚= 𝒆 𝒙 & 𝐲=𝐥𝐧 𝐱
3. A function and its inverse will be reflections of each other over the line 𝒚=𝒙.
𝒚= 𝒙 𝟑 & 𝐲= 𝟑 𝒙
𝒚= 𝒆 𝒙 & 𝐲=𝐥𝐧 𝐱

5. 5 Important Facts 𝑬𝒙. 𝒇 𝒙 =𝟑𝒙+𝟐 𝐠(𝐱)= 𝒙−𝟐 𝟑
4. To prove two functions are inverses, we use compositions:
𝒇 𝒈 𝒙 =𝒙 and 𝒈 𝒇 𝒙 =𝒙
𝑬𝒙. 𝒇 𝒙 =𝟑𝒙+𝟐
𝐠(𝐱)= 𝒙−𝟐 𝟑

6. 5 Important Facts
5. To write the inverse of a function, switch x and y, then solve for y.
Ex. 𝒇 𝒙 =−𝟑𝒙+𝟓

7. 1. Find the inverse, if it exists: 𝑓 𝑥 =𝑥+5

8. 2. Find the inverse, if it exists: 𝑓 𝑥 = 𝑥 3 +2

9. 3. Find the inverse, if it exists: 𝑓 𝑥 = 𝑥 2 −5

10. 4. Determine if the two functions are inverses: 𝑓 𝑥 =2−5𝑥 𝑔 𝑥 = 2−𝑥 5

11. 5. Determine if the two functions are inverses: 𝑓 𝑥 =4+6𝑥 𝑔 𝑥 = 6−𝑥 4

12. 6. Find each exact value: A. sin −1 (−1) B. cos −1 (− 2 2 )

13. 7. Find each exact value: A. tan −1 ( − 3 3 ) B. sin cos −1 (− 1 2 )