The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x)...
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Transcript of The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x)...
![Page 1: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/1.jpg)
The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f.
The composition of the function f with the function g is defined by
(f ○ g)(x) = f(g(x)).Two step process to find y = f(g(x)):
1. Find h = g(x).
2. Find y = f(h) = f(g(x))
Composite Functions
![Page 2: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/2.jpg)
A function, f, has an inverse function, g, if and only if
f ◦ g(x) = x and g ◦ f(x)) = x,for every x in domain of gand in the domain of f.
Definition of an Inverse Function
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Example 1
• Name two points on the inverse of the function t, when t(x) = x3 + 5x + 7
Plug in relatively easy numbers! (0,7), (1,13)
Now flip them!! (7,0) and (13,1)
![Page 4: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/4.jpg)
Is the inverse a function?
1. If the relation passes the vertical line test, it is a function.
2. If the relation passes the horizontal line test, its inverse is a function.
![Page 5: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/5.jpg)
1. Given the function y = f(x).
2. Interchange x and y.
3. Solve the result of Step 2 for y = g(x).
4. If y = g(x) is a function, then g(x) = f-1(x).
Finding the Inverse of a Function
![Page 6: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/6.jpg)
Example 2
• y = 18 – x2
• x = 18 – y2
• x – 18 = -y2
• -x + 18 = y2
![Page 7: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/7.jpg)
Find the inverse and then verify that the two functions are inverses of each other.
![Page 8: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/8.jpg)
• f ◦ g =
![Page 9: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/9.jpg)
• g ◦ f =
![Page 10: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/10.jpg)
Homework
Pages 212 - 213
2, 5 -13
![Page 11: The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f. The composition of the function.](https://reader035.fdocuments.in/reader035/viewer/2022072014/56649eaa5503460f94baf845/html5/thumbnails/11.jpg)
• Thanks to:
Dr. Claude S. MooreDanville Community
College