5.2 Inverse Function 2/22/2013

Function We defined a function as a relation between two sets, called the domain and range, such that for each x-value, there is exactly one y-value.

Determining if a graph is a function Vertical Line Test: The vertical line test is used to determine whether a graph is the graph of a function. Essentially, the test answers the question: is it possible to draw a vertical line that intersects the graph at more than one place? If yes, then the graph is not the graph of a function. If it is not possible, then the graph is the graph of a function.

Determine whether the graph is a function. No, vertical line crosses the graph at 2 places. Yes Yes Yes

Inverse Function A function and its inverse function can be described as the "DO" and the "UNDO" functions. A function takes a starting value, performs some operation on this value, and creates an output answer. The inverse function takes the output answer, performs some operation on it, and arrives back at the original function's starting value. Inverse function is written as 𝒇 −𝟏 𝒙 “f inverse of x” The graph of inverse function is the image of the original function reflected on the line y = x.

Here we have the function f(x) = 2x+3, written as a flow diagram: Example Here we have the function f(x) = 2x+3, written as a flow diagram: y = f(x) The Inverse Function just goes the other way: x So the inverse of 2x + 3 is 𝑦−3 2 𝑓(𝑥) 𝑓 −1 (𝑥)

x f (x) = 2x+ 3 x 𝒇 −𝟏 𝒙 = 𝒙−𝟑 𝟐 -2 -1 1 -1 1 3 5 -1 1 3 5 -2 -1 1 What do you notice about the 2 tables (The original function and it’s inverse)? The input of the original function is the output of the inverse and the output of the original function is the input of the inverse. The inverse of a function is the set of ordered pairs obtained by interchanging the domain(input) and range (output) values in the original function.

𝒇(𝒙)=𝟐𝒙+𝟑 𝒚=𝒙 𝒇 −𝟏 (𝒙)= 𝟏 𝟐 𝒙+ 𝟑 𝟐 Notice the graphs of 𝑓 𝑥 𝑎𝑛𝑑 𝑓 −1 𝑥 are mirror images of each other along y = x

One-to-One Function is a function in which the output corresponds to exactly 1 input. How do you determine if a function is a one-to-one function? It must pass both the Horizontal and Vertical Line test. Horizontal Line Test: Is a test that answers the question: is it possible to draw a horizontal line that intersects the graph at more than one place? If yes, then the graph is not a one to one function. If it is not possible, then the graph is a one-to-one function.

Determine whether the graph is a one-to-one function. Yes No, horizontal line crosses the graph at more than 1 place. No No

Assume that f is a one-to-one function. If 𝑓 3 =7 find 𝑓 −1 (7) If 𝑓 −4 =2 find 𝑓 −1 (2) If 𝑓 1 =6 and 𝑓 6 =10 find 𝑓 −1 (6) If 𝑓 −1 2 =5 find 𝑓 −1 (𝑓 5 ) 3 -4 1 5