Chapter 1 Functions and Their Graphs
Warm Up 1.6 A high-altitude spherical weather balloon expands as it rises due to the drop in atmospheric pressure. Suppose that the radius r increases at a constant rate of 0.03 inches per second and that r = 48 inches at time t = 0. Determine an equation that models the volume V of the balloon at time t and find the volume when t = 300 seconds. Note: Volume of a sphere. 2
1.6Inverse Functions Objectives: Find inverse functions numerically and algebraically. Verify that two functions are inverse functions of each other. Determine if functions are one-to-one. Use graphs of functions to decide whether functions have inverse functions. 3
Consider This … A highway crew is painting the center line on a road. Knowing the crew’s previous work record, it is possible to predict how much of the stripe the crew will have painted at any time during a normal 8-hour shift. It may also be possible to tell how long the crew has been working by how much stripe has been painted. 4
Mathematically … For the first case, the distance is dependent on time. For the second, time is dependent on the distance. 5
Inverses If a new relation is formed by interchanging the input and output variables in a given relation, the two relations are inverses of each other. If both relations are functions, they are called inverse functions. Notation: The inverse of f (x) is f -1 (x). 6
Inverse of a Function Numerically Let d be the number of miles a highway crew paints in an 8-hour shift. Let t be the number of hours the crew has been on the job. Find f (1), f (2), etc. Find g(1), g(2), etc. Inputs & outputs are interchanged. 7 t (hours) d (miles)
Graphs of the Functions 8 d = f (t)t = g(d)
Inverse of a Function Graphically When a function is graphed on the same set of axes as its inverse, we see that the function and its inverse are reflections across the line y = x. 9
Inverse of a Function Algebraically To find the inverse of a function algebraically: Exchange x -values and y -values. Solve for y. Call the original function f (x) and the inverse function f -1 (x). 10
Algebraic Inverse Example 11 Linear function for the function f. or Interchange variables. Solve for y in terms of x. Label function as f (x) and inverse as f -1 (x).
Composition of Inverse Functions The composition of a function f with its inverse f -1 results in the identity function. The functions f and f -1 “undo” one another. That is, and. 12
Domain and Range of Inverse Functions If we interchange the x -values and the y -values to create the inverse of a function, it must follow that: The domain of f is the range of f -1 and The range of f is the domain of f -1. Likewise, The domain of f -1 is the range of f and The range of f -1 is the domain of f. 13
Existence of an Inverse How do we know that a relation is a function? It passes the Vertical Line Test. Each input has only one output. For the function to have an inverse, Each output must have only one input. The function must be one-to-one. 14
One-to-One Functions A one-to-one function has exactly one y for each x and exactly one x for each y. If a function is one-to-one, it will pass the Horizontal Line Test. A function has an inverse if and only if it is one-to-one. If f is increasing on its entire domain or if it is decreasing on its entire domain, then f is one-to-one and, therefore, has an inverse. 15
Example 1 Given. a. Make a table of values for f (–2), f (–1), f (0), f (1), and f (2). From the numbers in the table, explain why you cannot find the value of x if f (x) = 2.5. How does this result indicate that we cannot find an inverse for f ? b. Plot the five points for f and connect with a smooth curve. Do the same for the inverse relation (on the same set of axes). How does the graph confirm that the inverse function for f cannot be found? c. Plot the line y = x on the same set of axes. What do you see? 16
Homework 1.6 Worksheet 1.6 # 6, 7, 9, 11, 17, 20, 21 – 24 (matching), 27, 32, 34, 39, 56, 57, 61, 62, 67, 68, 83, 84 17