![Page 1: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/1.jpg)
Section 8.2Inverse Functions
![Page 2: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/2.jpg)
• Consider the following two tables
• Are they both functions?• What are f-1 and g-1?
x f(x)
-2 10
-1 12
0 16
1 20
2 27
3 44
x g(x)
-5 -10
-2 -12
-1 -16
1 -16
2 -12
5 -10
![Page 3: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/3.jpg)
Definition of an Inverse Function• Suppose Q = f(t) is a function with the
property that each value of Q determines exactly one value of t. The f has an inverse function, f -1 and
If a function has an inverse, it is said to be invertible
).(ifonlyandif)(1 tfQtQf
![Page 4: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/4.jpg)
Graphs of Inverses• Consider the function 3)( xxf
![Page 5: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/5.jpg)
• The inverse is 3)( 21 xxf
![Page 6: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/6.jpg)
Plot of the two graphs together
![Page 7: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/7.jpg)
![Page 8: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/8.jpg)
Horizontal Line Test
• A function must be one-to-one in order to have an inverse (that is a function)
• A function is one-to-one if it passes the horizontal line test– A horizontal line may hit a graph in at most one
point
• We can restrict the domain of functions so their inverse exists– For example, if x ≥ 0, then we have an inverse for
3)( 2 xxg
![Page 9: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/9.jpg)
• Let’s talk about the following problems
)100(find1)(Given
)20(find23)(Given
)(find13)(Given
)(find13)(Given
15
13
1
13
kxxxk
hxxh
xgxg
xfxxf
x
x
![Page 10: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/10.jpg)
Property of Inverse Functions• If y = f(x) is an invertible function and y = f -1(x) is
its inverse, then
definedis)(
whichforofvaluesallfor))((
definedis)(
whichforofvaluesallfor))((
1
1
1
xf
xxxff
xf
xxxff
![Page 11: Section 8.2 Inverse Functions. Consider the following two tables Are they both functions? What are f -1 and g -1 ? xf(x)f(x) -210 12 016 120 227 344 xg(x)g(x)](https://reader035.vdocuments.net/reader035/viewer/2022080915/56649e2a5503460f94b17b42/html5/thumbnails/11.jpg)
• In your groups try problems 5, 15, and 25