Presentation on theme: "6.2 One-to-One Functions; Inverse Functions"— Presentation transcript:
1 6.2 One-to-One Functions; Inverse Functions MAT SPRING 20096.2 One-to-One Functions; Inverse FunctionsIn this section, we will study the following topics:One-to-one functionsFinding inverse functions algebraicallyFinding inverse functions graphicallyVerifying that two functions are inverses, algebraically and graphically
2 MAT SPRING 2009Inverse FunctionsExample*:Solution:
3 Inverse FunctionsNotice, for each of the compositions f(g(x)) and g(f(x)), the input was x and the output was x.That is because the functions f and g “undo” each other.In this example, the functions f and g are __________ ___________.
4 Definition of Inverse Functions MAT SPRING 2009Definition of Inverse FunctionsThe notation used to show the inverse of function f is f -1 (read “f-inverse”).Definition of Inverse Functionf -1 is the inverse of function ff(f -1(x)) = x for every x in the domain of f -1ANDf -1(f(x))=x for every x in the domain of fThe domain of f must be equal to the range of f -1, and the range of f must be equal to the domain of f -1.We will use this very definition to verify that two functions are inverses.
5 Watch out for confusing notation. MAT SPRING 2009Watch out for confusing notation.Always consider how it is used within the context of the problem.
6 MAT SPRING 2009ExampleUse the definition of inverse functions to verify algebraically thatare inverse functions.
7 A function f has an inverse if and only if f is ONE-TO-ONE. MAT SPRING 2009One-to-one FunctionsNot all functions have inverses. Therefore, the first step in any of these problems is to decide whether the function has an inverse.A function f has an inverse if and only if f is ONE-TO-ONE.A function is one-to-one if each y-value is assigned to only one x-value.
10 One-to-one FunctionsUsing the graph, it is easy to tell if the function is one-to-one. A function is one-to-one if its graph passes the H____________________ L_________ T__________ ; i.e. each horizontal line intersects the graph at most ONCE.one-to-oneNOT one-to-one
11 ExampleUse the graph to determine which of the functions are one-to-one.
12 Finding the Inverse Function Graphically To find the inverse GRAPHICALLY:Make a table of values for function f.Plot these points and sketch the graph of fMake a table of values for f -1 by switching the x and y-coordinates of the ordered pair solutions for fPlot these points and sketch the graph of f -1The graphs of inverse functions f and f -1 are reflections of one another in the line y = x.
13 Finding the Inverse Function Graphically (continued) Example:y=xf(x)=3x-5xf(x)-51-2234xf-1(x)-5-21243
15 Which of the following is the graph of the function below and its inverse?
16 Finding the Inverse Function Algebraically To find the inverse of a function ALGEBRAICALLY:First, use the Horizontal Line Test to decide whether f has an inverse function. If f is not 1-1, do not bother with the next steps.Replace f(x) with y.Switch x and ySolve the equation for y.Replace y with f -1(x).
17 Finding the Inverse Function Algebraically ExampleFind the inverse of each of the following functions, if possible.1.
18 Finding the Inverse Function Algebraically ExampleFind the inverse of each of the following functions, if possible.2.
19 Finding the Inverse Function Algebraically Example (cont.)3.
20 Finding the Inverse Function Algebraically Example (cont.)4.
21 Java AppletClick here to link to an applet demonstrating inverse functions.