Inverse Functions PDF

SH1902 Inverse Functions Steps in finding the inverse of a function 𝑓𝑓(𝑥𝑥): STEP 1. Let 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). STEP 2. Interchange the variables 𝑥𝑥 and 𝑦𝑦. STEP 3. Solve for 𝑦𝑦 in terms of 𝑥𝑥. STEP 4. Replace 𝑦𝑦 by 𝑓𝑓 −1 (𝑥𝑥). Let 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). Note t...

SH1902 Inverse Functions Steps in finding the inverse of a function 𝑓𝑓(𝑥𝑥): STEP 1. Let 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). STEP 2. Interchange the variables 𝑥𝑥 and 𝑦𝑦. STEP 3. Solve for 𝑦𝑦 in terms of 𝑥𝑥. STEP 4. Replace 𝑦𝑦 by 𝑓𝑓 −1 (𝑥𝑥). Let 𝑦𝑦 = 𝑓𝑓(𝑥𝑥). Note that 𝑓𝑓 −1 (𝑦𝑦) = 𝑥𝑥, that is, 𝑓𝑓 −1 undoes what 𝑓𝑓 does to a number 𝑥𝑥 within its domain. Example 1: Consider 𝑓𝑓(𝑥𝑥) = 4,050𝑥𝑥. STEP 1. Let 𝑦𝑦 = 4,050𝑥𝑥. STEP 2. Interchange the variables 𝑥𝑥 and 𝑦𝑦. 𝑥𝑥 = 4,050𝑦𝑦 STEP 3. Solve for 𝑦𝑦 in terms of 𝑥𝑥. 4,050𝑦𝑦 = 𝑥𝑥 𝑥𝑥 𝑦𝑦 = 4,050 STEP 4. Replace 𝑦𝑦 by 𝑓𝑓 −1 (𝑥𝑥). 𝑥𝑥 𝑓𝑓 −1 (𝑥𝑥) = 4,050 𝑥𝑥 Therefore, the inverse of 𝑓𝑓(𝑥𝑥) = 4,050𝑥𝑥 is 𝑓𝑓 −1 (𝑥𝑥) = 4,050. 7𝑥𝑥+5 Example 2: Consider 𝑔𝑔(𝑥𝑥) = 2. 7𝑥𝑥+5 STEP 1. Let 𝑦𝑦 = 2. STEP 2. Interchange the variables 𝑥𝑥 and 𝑦𝑦. 7𝑦𝑦+5 𝑥𝑥 = 2 STEP 3. Solve for 𝑦𝑦 in terms of 𝑥𝑥. 7𝑦𝑦+5 2 = 𝑥𝑥 7𝑦𝑦 + 5 = 2𝑥𝑥 7𝑦𝑦 = 2𝑥𝑥 − 5 2𝑥𝑥 − 5 𝑦𝑦 = 7 STEP 4. Replace 𝑦𝑦 by 𝑔𝑔−1 (𝑥𝑥). 2𝑥𝑥 − 5 𝑔𝑔−1 (𝑥𝑥) = 7 7𝑥𝑥+5 2𝑥𝑥−5 Therefore, the inverse of 𝑔𝑔(𝑥𝑥) = 2 is 𝑔𝑔−1 (𝑥𝑥) = 7. 03 Handout 1 *Property of STI  [email protected] Page 1 of 2  SH1902 Example 3: Consider ℎ(𝑥𝑥) = 4 − 3𝑥𝑥. STEP 1. Let 𝑦𝑦 = 4 − 3𝑥𝑥. STEP 2. Interchange the variables 𝑥𝑥 and 𝑦𝑦. 𝑥𝑥 = 4 − 3𝑦𝑦. STEP 3. Solve for 𝑦𝑦 in terms of 𝑥𝑥. 3𝑦𝑦 + 𝑥𝑥 = 4 3𝑦𝑦 = 4 − 𝑥𝑥 4 − 𝑥𝑥 𝑦𝑦 =. 3 STEP 4. Replace 𝑦𝑦 by ℎ−1 (𝑥𝑥). 4 − 𝑥𝑥 ℎ−1 (𝑥𝑥) = 3 4−𝑥𝑥 Therefore, the inverse of ℎ(𝑥𝑥) = 4 − 3𝑥𝑥 is ℎ−1 (𝑥𝑥) = 3. We know that if 𝑓𝑓 is invertible, and (𝑎𝑎, 𝑏𝑏) is a point on the graph of 𝑓𝑓, that is, 𝑓𝑓(𝑎𝑎) = 𝑏𝑏, then (𝑏𝑏, 𝑎𝑎) is a corresponding point on the graph of 𝑓𝑓 −1 (because 𝑓𝑓 −1 (𝑏𝑏) = 𝑓𝑓 −1 𝑓𝑓(𝑎𝑎) = 𝑎𝑎). This implies that the graph of 𝑓𝑓 −1 is the reflection of 𝑓𝑓 with respect to the line 𝑦𝑦 = 𝑥𝑥. If (𝑎𝑎, 𝑏𝑏) is a point in 𝑓𝑓, then (𝑏𝑏, 𝑎𝑎) is a point in 𝑓𝑓 −1. Source: https://www.desmos.com/calculator References Inverse functions. (n.d.). In Class Zone. Retrieved from http://www.classzone.com/eservices/home/pdf/teacher/LA207DBD.pdf Inverse functions. (2009). In Math Centre. Retrieved from http://www.mathcentre.ac.uk/resources/uploaded/mc-ty- inverse-2009-1.pdf How to find the inverse of a function. (2017). In Coolmath. Retrieved from http://www.coolmath.com/algebra/16-inverse- functions/05-how-to-find-the-inverse-of-a-function-01 Inverse Functions. (n.d.). In Math Is Fun. Retrieved from https://www.mathsisfun.com/sets/function-inverse.html Weisstein, E. (n.d.). Inverse function. In Wolfram Math World. Retrieved from http://mathworld.wolfram.com/InverseFunction.html Stapel, E. (2017). Finding the inverse of a function. In Purplemath. Retrieved from http://www.purplemath.com/modules/invrsfcn3.htm 03 Handout 1 *Property of STI  [email protected] Page 2 of 2