Composition and Inverses

1. Composition of Functions

Example:

Sociologists in Holland determine that the number of people y waiting in a water ride at an amusement park is given by

y = 1/50C + C + 2

where C is the temperature in degrees C.  The formula to convert Fahrenheit to Celsius C is given by

C = 5/9 F + 160/9

To get a function of F we compose the two function:

y(C(F)) = (1/50)[5/9F + 160/9]2 + (5/9F + 160/9) + 2

Exercises:

If

f(x) = 3x + 2

g(x) = 2x2 + 1

h(x) =

c(x) = 4

1.  Find f(g(x))

2. Find f(h(x))

3. Find f(f(x))

4. Find h(c(x))

5. c(f(g(h(x))))

2. 1-1 Functions
 Definition A function f(x) is 1-1 if            f(a) = f(b)  implies that            a = b

Example:

If

f(x) = 3x + 1

then

3a + 1 = 3b + 1

implies that

3a = 3b

hence

a = b

therefore f(x) is 1-1.

Example:

If

f(x) = x2

then

a2 = b2

implies that

a2 - b2  = 0

or that

(a - b)(a + b) = 0

hence

a = b or a = -b

For example

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

Hence f (x) is not 1-1.

3. Horizontal Line Test

If every horizontal line passes through f(x) at most once then f(x) is 1-1.

4. Inverse Functions

 Definition     A function g(x) is an inverse of f (x) if           f (g(x)) = g(f (x)) = x

Example:

The volume of a lake is modeled by the equation

V(t) = 1/125 h3

Show that the inverse is

h(N) = 5V1/3

We have

h(V(h)) = 5(1/125h3)1/3 = 5/5h = h

and

v(h(V)) = 1/125(5V1/3)3 = 1/125(125V) = V

5. Step by Step Process for Finding the Inverse:

1. Interchange the variables

2. Solve for y

3. Write in terms of f -1(x)

Example:

Find the inverse of

f (x) = y = 3x3 - 5

1. x =  3y3 - 5

2. x - 5 =  3y3 , (x - 5)/3 =  y3 , [(x - 5)/3]1/3

3. f -1(x) =  [(x - 5)/3]1/3

6. Graphing:

To graph an inverse we draw the y = x line and reflect the graph across this line.

http://mathcsjava.emporia.edu/~greenlar/Inverse/inverse.html

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