Inverse Functions 𝑓 −1 "𝑓 𝑖𝑛𝑣𝑒𝑟𝑠𝑒"

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: (−∞,∞)

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

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 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

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

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

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

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

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

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

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

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