We have seen that functions can be combined by addition , subtraction ,
multiplication , and division . We now investigate a new way to combine functions.
We call it composition of functions. Composition of functions can give us an
equation that allows us to eliminate steps when calculating . A composition
uses the output of one function as the input of another function.
Example 0.1 Your restaurant sells x liters of turtle soup. For each
liter
sold you must buy 2/3 of a pound of turtle meat. You pay $4.25 per pound for
turtle meat.
The function that ex presses how much turtle meat you need when selling x
liters of soup is f(x) = 2/3x.
The function that expresses how much you spend on turtle meat for each
pound of turtle meat is g(p) = (p)(4.25).
So if you sell 60 liters of soup in a day then you used f(60) = 2/3(60) = 40
pounds of turtle meat. So you spent g(40) = (40)(4.25) = 170 dollars for
turtle meat.
Using composition of functions we can create h(x) = (2/3x)(4.25). This
function tells us how much we spent on meat based on the number of liters sold.
This al lows us to get out final answer in one step instead of two.
h(60) = (2/3(60))(4.25) = 170 dollars spent on meat when 60 liters are sold.
We use the notation
Definition 0.1 The composition of the function f with g is denoted
and is defined by the equation
The domain of the composite function is the set of all x such that
1. x is in the domain of g and
2. g(x) is in the domain of f.
Example 0.2 Find
when f(x) = 7x + 1 and
Sometimes functions share a very special relationship, and are known as
inverses.
Think of inverse functions as functions that "undo" each other.
Example 0.3 The 20,000 gallon capacity swimming pool in your backyard
loses 50 gallons a day to evapo ration .
The function that describes this is f(x) = x - 50 where x is initial number
of gallons.
You put a hose continuously dribbling water into the pool at a rate of 50
gallons a day. This is described by g(x) = x + 50.
Notice that
.
In other
words, we added 50 gallons to the pool but 50 gallons evaporated so we ended
up with the same amount we started with. The action of putting the hose
in the pool and the of evaporation "undid" each other. Those actions are
inverses and their representative functions are inverses.
Definition 0.2 Let f and g be two functions such that
f(g(x)) = x for every x in the domain of g
and
g(f(x)) = x for every x in the domain of f
The function g is the inverse function of f, and is denoted g = f-1.
Example 0.4 Verifying Inverse Functions
Thus f and g are inverse functions.
Example 0.5 Verifying Inverse Functions
Thus f and g are not inverse functions.
Finding the inverse of a Function
1. Replace f(x) with y in the equation for f(x).
2. Inter change x and y .
3. Solve for y .
4. Replace y with f-1(x). Verify f-1(f(x)) = x and f(f-1(x)) = x.
Example 0.6 Find f-1(x)
Example 0.7 Find f-1(x)
Why did we specify x≥0?
Not all functions have inverses.
Consider f(x) = x^2
Remember that 
is
shorthand for 
So for each legitimate
input (x- value ) we get 2 outputs (y-values). This violates the condition of a
function.
How can we tell which functions have inverses?
Definition 0.3 Horizontal Line Test
A function f has an inverse that is a function, f-1, if there is no horizontal
line that intersects the graph of the function f at more than one point.
Note that there is a relationship between the graph of f
and its inverse, f-1.
If a point (a; b) is on the graph of f, then (b; a) is a point on the graph f-1.
This means that the graph f-1 is a re ection across the line y = x.
