inverse functions
DESCRIPTION
Inverse Functions. Objectives. Students will be able to find inverse functions and verify that two functions are inverse functions of each other. Students will be able to use the graph of a functions to determine whether functions have inverse functions. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/1.jpg)
Inverse Functions
![Page 2: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/2.jpg)
Objectives Students will be able to find inverse
functions and verify that two functions are inverse functions of each other.
Students will be able to use the graph of a functions to determine whether functions have inverse functions.
Students will be able to use the horizontal line test to determine if functions are one-to-one.
Find inverse functions algebraically.
![Page 3: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/3.jpg)
Definition of the Inverse Function
Let f and g be two functions such thatf (g(x)) = x for every x in the domain of gand g(f (x)) = x for every x in the domain of f.
The function g is the inverse of the function f, and denoted by f -1 (read “f-inverse”). Thus,
f ( f -1(x)) = x and f -1( f (x)) = x. The domain of f is equal to the range of f -1, and vice versa.
![Page 4: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/4.jpg)
ExampleShow that each function is the inverse of the other:
f (x) = 3x and g(x) = x/3.Solution To show that f and g are inverses of each other, we must show that f (g(x)) = x and g( f (x)) = x. We begin with f (g(x)).
f (x) = 3x
f (g(x)) = 3g(x) = 3(x/3) = x.
Next, we find g(f (x)).
g(x) = x/3
g(f (x)) = f (x)/3 = 3x/3 = x.
Notice how f -1 undoes the change produced by f.
![Page 5: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/5.jpg)
Graphing Inverses
• Inverses are symmetric about the line y=x
• To graph reverse the x and y coordinates.
• Use the symmetry around y = x
• Example; page 2117 - 118 #38, 16, 18
![Page 6: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/6.jpg)
The Horizontal Line Test For Inverse Functions
A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.
![Page 7: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/7.jpg)
Example
• Does f(x) = x2+3x-1 have an inverse function?
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
![Page 8: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/8.jpg)
ExampleSolution:
This graph does not pass the horizontal line test, so f(x) = x2+3x-1 does not have an inverse function.
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
![Page 9: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/9.jpg)
Finding the Inverse of a FunctionThe equation for the inverse of a function f can be found as
follows:1. Verify the function is one-to-one (HLT).2. Replace f (x) by y in the equation for f (x).3. Interchange x and y.4. Solve for y. If this equation does not define y as a function of
x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
5. If f has an inverse function, replace y in step 3 with f -1(x). We can verify our result by showing that f ( f -1(x)) = x and f -1( f (x)) = x.
![Page 10: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/10.jpg)
Find the inverse of f (x) = 6x + 3.Solution Step 1 Replace f (x) by y.
y = 6x + 3
Step 2 Interchange x and y.
x = 6y + 3 This is the inverse function.
Step 3 Solve for y.x - 3 = 6y Subtract 3 from both sides.
x - 3 = y6
Divide both sides by 6.
Step 4 Replace y by f -1(x).
x - 36
f -1(x) = Rename the function f -1(x).
Example
![Page 11: Inverse Functions](https://reader035.vdocuments.us/reader035/viewer/2022081516/56813929550346895da0d154/html5/thumbnails/11.jpg)
ExamplesExamples
Page 118 #58, 76Page 118 #58, 76
Groups: Pg 117: # 12, 28, 30, Groups: Pg 117: # 12, 28, 30, 39, 41, 6239, 41, 62
Homework: Pg 117: # 1 – 41 Homework: Pg 117: # 1 – 41 odd, 63, 67, 75odd, 63, 67, 75