Unit 1 Day 8 Inverse Functions F-BF.4: Find an inverse function and use this relationship to sole problems using tables, graphs, and equations.

Recall: Definition of a function In a function, each _______ can have only one ________ In other words, each ____ can only have one ____ Decide whether or not the relations below represent functions: input output x y YES or NO YES or NO YES or NO YES or NO YES or NO

Inverse Functions opposite “undo” Square root 𝑥 𝑥 𝑦 domain and range Inverse functions are ________________ or functions that ________________each other Think of the function 𝑓 𝑥 = 𝑥 2 . How do you “undo” squaring x? ______________________ 𝑥 2 and __________ are inverse functions In inverse functions, the __ and __ values are switched from the original function When x and y values switch places can you think of something else we have been learning about that might also switch? _____________________________________ When the x and y values switch, this results in a reflection over ____________ The inverse of the function f is labeled _______________. We read this as “f inverse.” Square root 𝑥 𝑥 𝑦 domain and range 𝑦=𝑥 𝑓 −1 (𝑥)

f −1 (x) For each table create a table that represents the inverse. Label the inverse correctly using function notation −4 −3 −2 −7 4 5 Does 𝑓(𝑥) represent a function? ____________ Does 𝑓 −1 (𝑥) represent a function? __________ Find 𝑓 0 : ______________________________________ What is 𝑓 −1 −3 ? _____________________________ What is 𝑓 −1 4 ? _____________________________ yes yes −7 −2 5

h −1 (x) For each table create a table that represents the inverse. Label the inverse correctly using function notation 9 −3 1 −1 1 1 yes Does ℎ(𝑥) represent a function? ____________ Does ℎ −1 (𝑥) represent a function? __________ Find ℎ 0 : ___________________________________ What is ℎ −1 9 ? _____________________________ What is ℎ −1 0 ? _____________________________ no −3

Find the y-value in 𝑓 −1 in the table, then look for x In the examples above, you were asked to evaluate the inverse function for a given input. Is there a pattern that you could use to evaluate the inverse of a function without creating an inverse table? __________________________________________________________________________________ If the point (−5, 3) is a point on 𝑓(𝑥), what point would be on 𝑓 −1 𝑥 ? _____________ If the point (8, 1) is a point on 𝑔(𝑥), what point would be on 𝑔 −1 𝑥 ? _______________ Find the y-value in 𝑓 −1 in the table, then look for x (3, −5) (1, 8) Use the table of 𝒇(𝒙) below to answer the following questions: 𝑓 −1 9 =__________ 𝑓 −1 (−2)= ________ −2 5

The function f(x) is shown on the graph The function f(x) is shown on the graph. Using the same approach, used with the tables, find the values requested: f −1 7 = _______ f −7 = _______ f −1 0 = _______ f 3 =_______ f −1 −3 = _______ f −1 −5 = _______ 5 −5 3 −7

Function Function Function Function Function Vertical Line Test To determine whether or not a graph represents a function we use the ______________________ If any vertical line touches the graph more than once, the graph is __________________ If any possible vertical line touches the graph only once, the graph is ________________ Determine whether or not the graphs below represent functions: Vertical line test not a function a function Function Function Function Function Function Not a Function Not a Function Not a Function Not a Function Not a Function

Inverse from Graphs To determine whether or not a functions inverse will also be a function use the ___________________ The same rules apply for an inverse function with the horizontal line test that apply for a function and the vertical line test If a function does not pass the horizontal line test, it will have an inverse on a _________________ All ______________ functions will have an inverse function with a restricted domain along the line of symmetry 𝐡𝐨𝐫𝐢𝐳𝐨𝐧𝐭𝐚𝐥 𝐥𝐢𝐧𝐞 𝐭𝐞𝐬𝐭 𝐫𝐞𝐬𝐭𝐫𝐢𝐜𝐭𝐞𝐝 𝐝𝐨𝐦𝐚𝐢𝐧 𝐪𝐮𝐚𝐝𝐫𝐚𝐭𝐢𝐜

For each graph below, determine whether or not the inverse would represent a function. If an inverse does not exist, use vertical lines to create a domain where an inverse would exist. Function Function Function Function Not a Function Not a Function Not a Function Not a Function

Inverse Functions 𝐥𝐢𝐧𝐞𝐚𝐫 𝐬𝐪𝐮𝐚𝐫𝐞 𝐫𝐨𝐨𝐭 𝐜𝐮𝐛𝐞 𝐫𝐨𝐨𝐭 The inverse of a linear function is always a ____________ function The inverse of a quadratic function is always a ______________ function The inverse of a cubic function is always a _____________ function 𝐥𝐢𝐧𝐞𝐚𝐫 𝐬𝐪𝐮𝐚𝐫𝐞 𝐫𝐨𝐨𝐭 𝐜𝐮𝐛𝐞 𝐫𝐨𝐨𝐭

Finding the inverse from an equation: Change the ____ to a ____ Switch the ____ and _____ Solve for _____ Use the notation ______ to represent your inverse 𝑓(𝑥) 𝑦 𝑥 𝑦 𝑦 𝑓 −1 (𝑥)

(1) Find the inverse of 𝑓 𝑥 =4𝑥−7 𝑦=4𝑥−7 𝑥=4𝑦−7 +7 +7 𝑥+7= 4𝑦 4 4 𝑓 −1 𝑥 = 𝑥+7 4

(2)Find the inverse of 𝑓 𝑥 =− 1 4 𝑥+8 𝑦=− 1 4 𝑥+8 𝑥=− 1 4 𝑦+8 −8 −8 −4 −4( ) 𝑥−8 = − 1 4 𝑦 𝑓 −1 𝑥 =−4𝑥+32

(3)Find the inverse of 𝑓 𝑥 =8 𝑥 2 −5 𝑦=8 𝑥 2 −5 𝑥=8 𝑦 2 −5 +5 +5 𝑥+5= 8 𝑦 2 8 8 𝑥+5 8 = y 2 𝑓 −1 𝑥 =± 𝑥+5 8 ,𝑥≥−5

(4)Find the inverse of 𝑓 𝑥 = 𝑥+1 5 , 𝑥≥−1 𝑦= 𝑥+1 5 𝑥 = 𝑦+1 5 5 5( ) 2 2 𝑦+1 5𝑥= 25 𝑥 2 = 𝑦+1 −1 −1 𝑓 −1 𝑥 =25 𝑥 2 −1

(5)Find the inverse of 𝑓 𝑥 = 𝑥 −4 𝑦= 𝑥 −4 𝑥= 𝑦 −4 +4 +4 𝑥+4 = 2 2 𝑦 𝑓 −1 𝑥 = 𝑥+4 2