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AP Calculus AB. Day 6 Section 5.3. Inverse Functions. If f ( g (x)) = x and g ( f (x)) = x then f (x) and g (x) are inverses. Domain of f (x) = Range of f -1 (x) Range of f (x) = Domain of f -1 (x) Inverses are symmetric about y = x. - PowerPoint PPT Presentation
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04/19/23 Perkins
AP Calculus AB
Day 6Section 5.3
Inverse Functions• If f(g(x)) = x and g(f(x)) = x then f(x) and g(x) are inverses.
• Domain of f(x) = Range of f -1(x) Range of f(x) = Domain of f -1(x)
• Inverses are symmetric about y = x.
• A function can only have an inverse if it is 1-to-1.2 ways to check 1-to-1:
a. horizontal line testb. is it always inc or dec?
Note: if a function isn’t 1-to-1 we can change its domain to make it 1-to-1.
4
2
-2
-4
(b,a)
(a,b)
1. Show that these functions are inverses:
29 if 0, 9f x x x g x x You could…a. Graph each and show symmetry about y = x.b. Show that both f(g(x)) = x and g(f(x)) = x.c. Find the inverse of one of the functions and compare.
To find an inverse:• Swap x & y.• Solve for y.• Domain of f -1(x) = Range of f(x).
29y x 29x y
29x y 29 x y
9 x y
: 0,D : ,9R
: ,9D : 0,R
19 x f x
These functions are inverses.
Can these functions have inverses?
33. y x23
dyx
dx
This derivative is always positive, so y is always increasing.
This function can have an inverse.
24. y x
2dy
xdx
This derivative changes signs, so y increases and decreases.
This function can’t have an inverse.
2. 60
40
20
-20
-40
-60
-10 10 This function can’t have an inverse.
10
8
6
4
2
-2
-4
-5 5
6
4
2
-2
-4
-6
-5 5
…unless we limit its domain to all positives or all negatives.
5. Find the inverse of
This function isn’t 1-to-1.
2 3y x
2 3x y 2 2 3x y
2 3 2x y 2 3
2
xy
Limit its domain.
3: ,
2D
: 0,R
3: ,
2R
: 0,D
2
1 3
2
xf x
Perkins
AP Calculus AB
Day 6Section 5.3
Inverse Functions• If f(g(x)) = x and g(f(x)) = x then f(x) and g(x) are inverses.
• Domain of f(x) = Range of f -1(x) Range of f(x) = Domain of f -1(x)
• Inverses are symmetric about y = x.
• A function can only have an inverse if it is 1-to-1.2 ways to check 1-to-1:
a.b.
4
2
-2
-4
(b,a)
(a,b)
1. Show that these functions are inverses:
29 if 0, 9f x x x g x x
Can these functions have inverses?
33. y x 24. y x
2. 60
40
20
-20
-40
-60
-10 10
5. Find the inverse of 2 3y x
