Composition and Inverses
I.
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 = 1/50C2 + 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
A.
B.
C.
D.
E.
II.
Find f(g(x))
Find f(h(x))
Find f(f(x))
Find h(c(x))
c(f(g(h(x))))
1-1 Functions
Definition
A function f(x) is 1-1 if
f(a) = f(b)
implies that
a=b
III.
Example:
If
f(x) = 3x + 1
then
3a + 1 = 3b + 1
implies that
IV.
3a = 3b
hence
a=b
therefore f(x) is 1-1.
V.
Example:
If
f(x) = x2
then
a2 = b2
implies that
a2 - b2 = 0
or that
(a - b)(a + b) = 0
VI.
hence
a = b or a = -b
For example
f (2) = f (-2) = 4
Hence f (x) is not 1-1.
VII.
VIII.
Horizontal Line Test
If every horizontal line passes through f(x) at most once then f(x) is 1-1.
IX.
Inverse Functions
Definition
A function g(x) is an inverse of f (x) if
f (g(x)) = g(f (x)) = x
X.
Example:
XI.
The volume of a lake is modeled by the equation
V(t) = 1/125 h3
XII.
XIII.
XIV.
Show that the inverse is
h(N) = 5V1/3
We have
h(V(h)) = 5(1/125h3)1/3 = 5/5h = h
XV.
and
v(h(V)) = 1/125(5V1/3)3 = 1/125(125V) = V
XVI.
XVII.
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
4. x = 3y3 - 5
5. x - 5 = 3y3 , (x - 5)/3 = y3 , [(x - 5)/3]1/3
6. f -1(x) = [(x - 5)/3]1/3
XVIII.
Graphing:
To graph an inverse we draw the y = x line and reflect the graph across this line.