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NotesStarts from 1.6

# NotesStarts from 1.6 - Section 1.6 Skip 1.4,1.5 Warning not...

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Section 1.6 Skip 1.4,1.5 Warning: not all functions have inverses Given a function f(X) if an inverse exiss, the the inverse function is defined by f -1 (Y) = x Where x is given by the solution to the equation y=f(X) Example: F(x) = x 3 what is f -1 (8)? f -1 (8)=x x3=8 x-2 f -1 (8)=2 generally, for this f, f-1(x)= x cancellation: f -1 (f(x))=x (valid for all x in domain of f) F(f -1 (y))=y Valid for y in range of f Example: x 3 =x Harder Example: f(x)=3x-3π+4+sin(2x) what is f -1 (4)? 4= 3x-3π+4+sin(2x) x=π becasue of sin (2x), just guessing!

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f -1 (4)= π Definition: afunction is one-to-one if it passes the horizontal line test. Ot equivalently, if it never achieves the value twice. Or symbolically: f(x 1 ) f(x 2 ) whenever x 1 x 2 . Adorithm for algebraically solving for the inverse 1- write y = f(x) 2- solve for x in terms of y (if possible) x = f -1 (y) 3- swap letters x y, y=f -1 (x) Example: F(x)= + x2x 1 1- Y= + x2x 1 2- 2xy+1=x x = - y1 2y 3- f -1 (y)= - y1 2y f -1 (x) =
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NotesStarts from 1.6 - Section 1.6 Skip 1.4,1.5 Warning not...

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