Section 3.7 - Inverse Functions

One-To-One Functions A one-to-one function is a function where each output corresponds to exactly one input. Algebraically, a function 𝑓 is one-to-one if 𝑓 𝑎 =𝑓 𝑏 ⇒𝑎=𝑏

One-To-One Functions Given the function f, prove that f is one-to-one using the definition of a one-to-one function. Since 𝑓 𝑎 =𝑓 𝑏 ⇒𝑎=𝑏, the function is 1 - 1. Since 𝑓 𝑎 =𝑓 𝑏 ⇒𝑎=𝑏, the function is 1 - 1.

One-To-One Functions Horizontal Line Test To determine if a function is 1-1 graphically, we use the horizontal line test. Horizontal Line Test Each horizontal line passes through the graph of a 1-1 function no more than once. Given a graph, we can say that it is a 1-1 function if it passes BOTH the vertical and horizontal line tests.

One-To-One Functions Using the horizontal line test, determine whether the function is 1-1.

One-To-One Functions Using the horizontal line test, determine whether the function is 1-1. It is possible to restrict the domain so that you have a 1-1 function.

Inverse Functions If a function is 1-1, we say it is “invertible” and has an inverse that is a function. We can think of an inverse function as a FUNCTION THAT WILL “UNDO” whatever the function does! We denote the inverse function of 𝑓 as 𝑓 −1 (𝑥).

Graphs of functions & Inverse Functions 5. Given the graph of a 1-1 function, graph the inverse function. Dom 𝑓: [0, 4] Range 𝑓: [-5, 11] Dom 𝑓 −1 : [−5, 11] Range 𝑓 −1 : [0, 4] The graph of the inverse is the reflection of the graph across the line y = x.

Graphs of functions & Inverse Functions 6. Graph the inverse of 𝑓 𝑥 = 2 −𝑥 , if it exists. Since this function passes the horizontal line test, it is 1 -1, therefore it has an inverse function. The point (0, 1) is on the graph of the function. The point (1,0) is on the graph of the inverse. The point (-3, 8) is on the graph of the function. The point (8, -3) is on the graph of the inverse. The positive x-axis is an asymptote on the graph of the function. The positive y-axis is an asymptote on the graph of the inverse.

Inverse Functions 7. Find the inverse of 𝑓 𝑥 , if it exists. Does an inverse exist? Graph the function and determine if it passes the horizontal line test. The function is 1 – 1, therefore the inverse exists.

Inverse Functions 3. Solve for 𝑦. 1. Replace 𝑓(𝑥) with 𝑦. 2. Interchange 𝑥 and 𝑦. 4. Change back to function notation.

Inverse Functions

Inverse Functions Does an inverse exist? Graph the function and determine if it passes the horizontal line test. The function is 1 – 1, therefore the inverse exists.

Inverse Functions 1. Replace 𝑓(𝑥) with 𝑦. 4. Change back to function notation. 2. Interchange 𝑥 and 𝑦. 3. Solve for 𝑦. Since the domain of g is x ≥ 0, we want the range of the inverse function to be greater than or equal to 0. We choose only the positive solution to y.

PROPERTIES OF INVERSE FUNCTIONS If (𝑥, 𝑦) is a point on the graph of the function, then the point (𝑦, 𝑥) is on the graph of the inverse. The domain of the function is the range of the inverse function. The range of the function is the domain of the inverse function. The graphs of a function and its inverse are reflections across the line y = x. The composition of a function with its inverse is the identity function. 𝑓 −1 ∘𝑓 𝑥 = 𝑓∘ 𝑓 −1 𝑥 =𝑥 𝑓 −1 𝑓(𝑥) =𝑥 𝑓 𝑓 −1 (𝑥) =𝑥

Composition of Inverse Functions For the function 𝑓 , use composition of functions to show that 𝑓 −1 is as given. Since we get x in both cases, the functions are inverses of each other.

Inverse Functions

Bus Chartering 10. Bus Chartering. An organization determines that the cost per person of chartering a bus is given by the formula 𝐶 𝑥 = 100+5𝑥 𝑥 , where 𝑥 is the number of people in the group and 𝐶(𝑥) is in dollars. Determine 𝐶 −1 (𝑥) and explain what it means. Does an inverse exist? Domain and Range of 𝐶 and 𝐶 −1 : 𝐶: (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒, 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑝𝑒𝑟𝑠𝑜𝑛) 𝐶 −1 : (𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑝𝑒𝑟𝑠𝑜𝑛, 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑒𝑜𝑝𝑙𝑒)