Section 3.1 Inverse Functions
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1 19 February 2016
2 First Example Consider functions and f (x) = 9 5 x + 32 g(x) = 5 9( x 32 ).
3 First Example Continued Here is a table of some points for f and g:
4 First Example Continued Here is a table of some points for f and g: f (x) ( 20, 4) (0, 32) (20, 68) (35, 95) ( C, F) g(x) ( 4, 20) (32, 0) (68, 20) (95, 35) ( F, C)
5 First Example Continued Here is a table of some points for f and g: f (x) ( 20, 4) (0, 32) (20, 68) (35, 95) ( C, F) g(x) ( 4, 20) (32, 0) (68, 20) (95, 35) ( F, C) Note: If (a, b) is a point associated with f, then (b, a) is a point associated with g.
6 Composition: f g and g f For the functions and f (x) = 9 5 x + 32 g(x) = 5 9( x 32 ), calculate f g and g f.
7 Inverse Function f 1 Definition: Given a function f, the inverse function of f, written f 1, is a function such that f (f 1 (x)) = x for all x in the domain of f 1,
8 Inverse Function f 1 Definition: Given a function f, the inverse function of f, written f 1, is a function such that f (f 1 (x)) = x for all x in the domain of f 1, and f 1 (f (x)) = x for all x in the domain of f.
9 Inverse Function f 1 Definition: Given a function f, the inverse function of f, written f 1, is a function such that f (f 1 (x)) = x for all x in the domain of f 1, and f 1 (f (x)) = x for all x in the domain of f. Also, if (a, b) is a point associated with f,
10 Inverse Function f 1 Definition: Given a function f, the inverse function of f, written f 1, is a function such that f (f 1 (x)) = x for all x in the domain of f 1, and f 1 (f (x)) = x for all x in the domain of f. Also, if (a, b) is a point associated with f, then (b, a) is a point associated with f 1 : f (a) = b a = f 1 (b).
11 Symmetry Between f and f 1 For f (x) = x 3, f 1 (x) = 3 x
12 Example with Straight Lines Show that and f (x) = 3x 6 g(x) = 1 3 x + 2 are inverses of each other.
13 Graphing Inverse Pairs Graph the inverse of the given function on the same axes: 4 4 (a) (b) (c)
14 One-to-One Functions Definition:
15 One-to-One Functions Definition: A function f is a one-to-one function
16 One-to-One Functions Definition: A function f is a one-to-one function if and only if
17 One-to-One Functions Definition: A function f is a one-to-one function if and only if we have f (x 1 ) f (x 2 ) whenever x 1 x 2.
18 One-to-One Functions Definition: A function f is a one-to-one function if and only if we have f (x 1 ) f (x 2 ) whenever x 1 x 2. This is equivalent to the Horizontal Line Test:
19 One-to-One Functions Definition: A function f is a one-to-one function if and only if we have f (x 1 ) f (x 2 ) whenever x 1 x 2. This is equivalent to the Horizontal Line Test: A function f is a one-to-one function if no horizontal line intersects its graph more than once.
20 One-to-One? Is f (x) = x 3 one-to-one?
21 One-to-One? Is f (x) = x 3 one-to-one? Is f (x) = x 2 one-to-one?
22 One-to-One? Is f (x) = x 3 one-to-one? Is f (x) = x 2 one-to-one? Is f (x) = x 2 one-to-one if x 0?
23 Theorem: A function f has an inverse f 1 if and only if f is a one-to-one function.
24 Is f one-to-one? 1. f (x) = 12x f (x) = x 3 4x 3. f (x) = x f (x) = { x + 1 : x 0 x : x > 0.
25 Finding an Inverse:
26 Finding an Inverse: 1. f (x) = 12x + 7
27 Finding an Inverse: 1. f (x) = 12x f (x) = x + 5, x 5
28 Finding an Inverse: 1. f (x) = 12x f (x) = x + 5, x 5 3. f (x) = 3x + 7 x 2, x 2
29 Finding an Inverse: 1. f (x) = 12x f (x) = x + 5, x 5 3. f (x) = 3x + 7 x 2, x 2 4. f (x) = x 2 + 4x + 3, x 2
30 Miscellaneous Problems: 1. If f (x) = x 2, find f 1 (0). 2. If f 1 (x) = 3x 2, find f (2). 3. Find values on which f (x) = x 2 2x + 3 is a one-to-one function. 4. Find f 1 and any restrictions if f (x) = 2x x 1, x 1.
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