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1.3

NEW FUNCTIONS FROM OLD

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2( )A f r r Area of oil spilled from a tanker grows with time:

Radius is also a function of time: ( ) 1r g t t

Area given as a function of time:2 2

2 2

(1 )

( ( )) ( ( )) (1 )

A r t

A f g t g t t

After 5 hours, t=5, 2 2( (5)) ( (5)) (1 5) 36A f g g square feet

+

4

3 27

3 3 27

( ) 1 ( )

( ( )) (1 ) (1 )

Let g t t and f t t

f g t f t t

Example 2: Express h(t) = (1 + t3)27 as a composition of two functions:

3 27

3 3 27

( ) ( ) (1 )

( ( )) ( ) (1 )

Let g t t and f t t

f g t f t t

Or

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For example, f(x) = x4 + x2 is even because f(-x) = (-x)4 + (-x)2 = x4 + x2 = f(x) and f(x) = x3 + x is old because f(-x) = (-x)3 + (-x) = -x3 + (-x) = -(x3 + x ) = -f(x)

2 13, ( ) ( ) .xSimilarly g x e is even and h x x is odd

1(4000) 629.98 (629.98) 4000f and wewrite f

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Independent variable for f is the dependent variable for f-1 .

The domain for f is the range for f-1 and the range for f is the domain for f-1

-1 1

f(-2) = 4 and f(2) = 4

But f (4) 2 (4) 2and f NOT A FUNCTION

Function f must be one-to-one for f-1 to exist.

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Example: Consider the function y = x3 , find f-1 .

Solution: To find the inverse, we solve for x, giving x = y1/3 .

So the inverse function is f-1(y) = y1/3 or if we want to call the independent x,

f-1(x) = x1/3

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