Inverses
| Site: | Farwell |
| Course: | Michigan Algebra I Sept. 2012 |
| Book: | Inverses |
| Printed by: | Guest user |
| Date: | Thursday, November 21, 2024, 9:31 PM |
Description
Inverses
Introduction
- If a function answers the question: "Alice worked this long; how much money has she made?" Then its inverse answers the question: "Alice made this much money; how long did she work?"
- If a function answers the question: "How many hours of music fit on 12 CDs?" Then its inverse answers the question: "How many CDs do you need for 3 hours of music?"
If a function is modeled by the rule, 
, then the inverse function is modeled by 
. Since multiplication and division are inverse operations, they will create inverse functions.
The function, f -1(x), is defined as the inverse function of f(x) if it consistently reverses the process of f(x). That is, if f(x) turns a into b, then f -1(x) must turn b into a. f -1(x) is not an exponent. It is meant to be an inverse function, not a reciprocal.
The inverse has all the same points as the original function, except that the x and y-coordinates have been reversed. If the point (3,2) is a point in the original function then the point (2,3) will be a point of the inverse function.
It is possible to find the inverse of a set of points, a table, a graph or an equation. The set of points and the table work the same way. All three methods for finding inverses of functions will be discussed here.
Set of Points
Whenever a function is given as a list of ordered pairs, simply switch the x and y-values to create an inverse.
Example Find the inverse of 
, and determine if the inverse is a function.
Step 1. Switch the x and y-coordinates.
Step 2. Determine if the inverse is a function.
Since every input has exactly one output, the inverse is a function. This function is called "one-to-one" which means every x and y value is used only once.Equation
Example Find the inverse of the equation: 
Step 1. Substitute a y for the f(x).
Step 2. Switch the x and y-variables.
Linear Graph
Example Graph the inverse of the equation y = 2x + 3.
Step 2. Graph the equation y = x.
Remember, not all graphs produce an inverse that is also a function. This is true with quadratic graphs.
Quadratic Functions
The horizontal line test states: if any horizontal line intersects the graph of a function more than once, then the function does not have an inverse that is also a function.
Below are the graphs of y = x2 and x = y2:
The graph of y = x 2 does not pass the horizontal line test so its inverse, x = y 2 is not a function. The graph of x = y 2 does not pass the vertical line test, so it verifies that the horizontal line test on the original function works. Therefore, the graph of f(x) = x 2 shows that Quadratic functions have inverse relations, but their inverses are not functions.
Video Lesson
Practice
Answer Key
Sources
Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project
Holt, Rinehart & Winston, "Inverse Functions." http://my.hrw.com/math06_07/nsmedia/lesson_videos/alg2/player.html?contentSrc=6481/6481.xml (accessed 8/22/2010).
"How to Find an Inverse Function." http://www.helpalgebra.com/articles/ inverseofafunction.htm (accessed 8/07/2010).
Regents Exam Prep Center, "Definition of Inverses." http://www.regentsprep.org/Regents/math/algtrig/ATP8/inverselesson.htm (accessed 08/07/2010).