Chapter 5: Exponential and Logarithmic Functions 5.1 Inverse Functions Goals/Objectives: Analyze 1-1 functions and inverses Evaluate functions that are inverses Analytically determine if two functions are inverses Synthesize a function’s inverse Restrict the domain of a non 1-1 function so that it becomes 1-1.

How do we read this? a and b come from the domain so what types of values are domain values? a cannot equal b. So they have to be different x values. f(a) cannot equal f(b). So the y values cannot be equal. So for different x values you have to have different y-vales (and by different we mean that for every x value, there is only 1 unique y value that goes with it. The word implies means “if and only if” which from geometry we know is a biconditional. Which means that this statement can be read from the a cannot equal b side and gain the f(a) cannot equal f(b) side, or start with the f(b) cannot equal f(a) side and gain the a cannot equal b side. 5.1 One-to-One Functions A function f is a one-to-one function if, for elements a and b from the domain of f, a  b implies f (a)  f (b).

5.1 Horizontal Line Test Example Use the horizontal line test to determine whether the graphs are graphs of one-to-one functions. (a) (b) If every horizontal line intersects the graph of a function at no more than one point, then the function is one-to-one. Not one-to-one One-to-one

Only functions that are one-to-one (1-1) have inverses. In order to be a 1-1 function it first must be a function so it must pass the vertical line test before we even try the horizontal line test Inverse functions are two functions that undo one another. If I put an x value into one function and return a y value. Then take that y value and let it be an input into a second function, the second function will then produce an output that was IDENTICAL to the first x you started with

This example shows the previous idea, first try inputting x=2 are inverse functions of each other. In order to “show” or “prove” they are inverses we have to show this for all possible x values. Which is impossible. So we have to use algebra techniques. (See next slide)

Let f be a one-to-one function Let f be a one-to-one function. Then, g is the inverse function of f and f is the inverse of g if

Determine algebraically whether   f (x) = 3x – 2 and g(x) = (x + 2)/3 are inverses of each other.  

Determine algebraically whether  f (x) = 3x – 2 and  g(x) = (1/3)x + 2 are inverses of each other.

The Graph of f -1(x) f and f -1(x) are inverse functions, and f (a) = b for real numbers a and b. Then f -1(b) = a. If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1. If a function is one-to-one, the graph of its inverse f -1(x) is a reflection of the graph of f across the line y = x.

5.1 Finding an Equation for the Inverse Function Notation for the inverse function f -1 is read “f-inverse”. By the definition of inverse function, the domain of f equals the range of f -1, and the range of f equals the domain of f -1.

5.1 Finding an Equation for the Inverse Function Finding the Equation of the Inverse of y = f(x) For a one-to-one function f defined by an equation y = f(x), find the defining equation of the inverse as follows. (Any restrictions on x and y should be considered.) 1. Interchange x and y. 2. Solve for y. 3. Replace y with f -1(x).

5.1 Example of Finding f -1(x) Example Find the inverse, if it exists, of Solution Write f (x) = y. Interchange x and y. Solve for y. Replace y with f -1(x).

5.1 Finding the Inverse of a Function with a Restricted Domain Example Let Solution Notice that the domain of f is restricted to [–5,), and its range is [0, ). It is one-to-one and thus has an inverse. The range of f is the domain of f -1, so its inverse is

5.1 Important Facts About Inverses If f is one-to-one, then f -1 exists. The domain of f is the range of f -1, and the range of f is the domain of f -1. If the point (a,b) is on the graph of f, then the point (b,a) is on the graph of f -1, so the graphs of f and f -1 are reflections of each other across the line y = x. Goals/Objectives: Analyze 1-1 functions and inverses Evaluate functions that are inverses Analytically determine if two functions are inverses Synthesize a function’s inverse Restrict the domain of a non 1-1 function so that it becomes 1-1.

5.1 Application of Inverse Functions Example Use the one-to-one function f (x) = 3x + 1 and the numerical values in the table to code the message BE VERY CAREFUL. A 1 F 6 K 11 P 16 U 21 B 2 G 7 L 12 Q 17 V 22 C 3 H 8 M 13 R 18 W 23 D 4 I 9 N 14 S 19 X 24 E 5 J 10 O 15 T 20 Y 25 Z 26 Solution BE VERY CAREFUL would be encoded as 7 16 67 16 55 76 10 4 55 16 19 64 37 because B corresponds to 2, and f (2) = 3(2) + 1 = 7, and so on.