**Functions and Their
Compositions**

**Damarrio C. Holloway**

In this essay, we will look at the effects of functions and their compositions with other functions. We will make conjectures of what type of functions we should have when composing certain types. We will examine these graphs on the Graphing Calculator program, but the students can graph and discuss them at their seats with individual graphing calculators.

LetÕs begin by briefly defining composite functions:

In mathematics, a composite
function, formed by the composition of one function on another, represents the
application of the former to the result of the application of the latter to the
argument of the composite. In other words, composition of functions is the
process of combining two functions where one function is performed first (in
which its range is the domain of the composition) and the result of which is
substituted in place of each x in the other function.

LetÕs look at some specific types
of functions. Here we will compose
and graph these functions and hopefully make inferences on the results.

__Constant vs. Constant__

Here we have two linear equations f(x) = 3 and g(x) = -5

1. What would the equations for the compositions of f(g(x)) and
g(f(x)) look like on the graph?
What do they mean?

2. What general statement can we make about the composition of
two functions onto each other?

__Linear vs. Linear__

Here we have two linear equations f(x) = 3x – 5 and g(x)
= -x + 7.

Our composition functions are f(g(x)) = f(-x + 7) = -3x + 6 and g(f(x)) = g(3x – 5) = -3x + 12. The
linear functions, when composed onto each other result in linear
functions. Also, these
compositions are parallel.

In a general case with, f(x) = ax
+ b and g(x) = cx + d, what do you notice about the slopes? Do you think this
always happens?

*f(g(x))* = *f*(c**x** + d) *g(f(x))* = *g*(a**x** + b)

= a(c**x** + d) + b =
c(a**x** + b) + d

= ac**x** + (ad + b) =
ca**x** + (cb + d)

What do you suppose will happen if
we compose a linear with a constant?

Now letÕs look at __Linear vs.
Quadratic__

Lets look at the graphs with f(x) = 2x + 1 and g(x) = . In our last example, we both of our
composition functions resulted in linear equations with the same slope. LetÕs see what type of functions will
be generated with:

As we composed a quadratic with a quadratic, we yield another quadratic function. Again the question is, does it always happen? If so provide a general explanation. If not, provide a counter example.

__Quadratic vs. Quadratic__

LetÕs look at the functions f(x) = - and g(x) =

In the Linear vs. Linear composition, we were able to yield a linear composite function. In our Quadratic vs. Quadratic composite function, our result is a polynomial with leading coefficient of 4, with 4 roots instead of 2.

f(g(x)) = f() = =

g(f(x)) = g(-)
= =

A. In groups, explain how the minimums and maximums of the original functions f(x) and g(x) relate to their composites f(g(x)) and g(f(x)).

B. Give a conjecture why the functions resulted in polynomials of root 4.

C. Write or explain a general equation when polynomials of root n > 2 are composed onto polynomials of root n > 2.

D. Also, what happens when you compose a constant with a quadratic or a quadratic with a constant? Graph these functions and write a general statement to find the composite function of quadratics onto constants.

__Inverses__

When composing Linear vs. Linear, a linear function was the result. When we composed the inverses of linear functions, we are again composing a linear function onto a linear function and , which should yield linear compositions. But what happens if we try to find the inverse of the composition of functions? In other words, find .

In our graphs we have the functions f(x) = 3x – 5 and g(x) = -x + 7 on the left and in blue and in green.

LetÕs first find the inverse of f(g(x)) and g(f(x)):

f(g(x)) = f(-x + 7) = -3x + 6 and g(f(x)) = g(3x – 5) = -3x + 12

1. Based on the graphs of the inverses of f(x) and g(x), what should the inverses of f(g(x)) and g(f(x)) look like? Graph to check your answer.

2. What does the graph of and look like? What can you say about inverses and their compositions?

LetÕs look at some ways to use and interpret graphs of composite functions in real-life situations.

Problem:

You make a purchase at a local
hardware store, but what youÕve bought is too big to take home in your
car. For a small fee, you arrange
to have the hardware store deliver your purchase for you. You pay for your purchase, plus the
sales taxes, plus the fee. The
taxes are 7.5% and the fee is $20.

a. Write a function t(x) for the total, after taxes, on
purchase amount x. Write another
function f(x) for the total, including the delivery fee, on purchase amount
x.

b. Calculate and interpret f(g(x)) and t(f(x)). Which results in a lower cost to you?

b. Suppose taxes, by law, are not to be charged on delivery
fees. Which composite function
must then be used?

Is the best composition function f((g(x)) or (f o g)(x).

f(x) = 1.075x in green g(x) = x + 20 in orange

f(g(x)) = (f o g)(x) = 1.075 (x +20) in red

g(f(x)) = (g o f)(x) = (1.075x) + 20 in blue

This sort of calculation actually
comes up in "real life", and is used for programming the cash
registers. And this is why there is a separate button on the register for
delivery fees and why they're not rung up as just another purchase.

1. The
taxes are 7.5%, so the tax function is given by *t(x) = 1.075x*

The delivery fee is fixed, so the
purchase amount is irrelevant.

The
fee function is given by *f(x) = x +20*

2. f(t(x) ¸ This function represents the total price of the furniture with an added
delivery fee which is not taxed.

f(x
+ 0.075x) = x + 0.075x + 20

= 1.075x + 20

t(f(x) ¸ This function represents the total amount I would pay if
the delivery fee is added to the purchase and then taxed.

t(x
+ 20) = x + 20 + 0.075(x + 20)

= 1.075x + 21.50

Though the two equations have the
same rate of change (slope), the f(t(x)) will give me the lowest price because
it has the lower starting value (y-intercept).

LetÕs take a graphical approach to
this portion of the problem:

By examining the graphs of each of
these functions f(x), t(x), f(t(x)), t(f(x)), the slopes of these linear
functions is one of our primary concerns when answering this question. The steeper the line, the greater rate
of change, in this case, the steeper the line, the more the customer will pay
in the long run. Another key
component is the y-intercepts of each of our lines. We can see that three of our functions t(x), f(t(x)), and
t(f(x)) all have the same slope 1.075, so the starting points of these graphs
(where they cross the y-axis is where they make a purchase) helps make the
decision of the lowest cost.

Here is a similar problem:

Your computer's screen saver is an
expanding circle. The circle starts as a dot in the middle of the screen and
expands outward, changing colors as it grows. With a twenty-one inch screen,
you have a viewing area with a 10-inch radius (measured from the center
diagonally down to a corner). The circle reaches the corners in four seconds.
Express the area of the circle (discounting the area cut off by the edges of
the viewing area) as a function of time t in seconds.

With the functions r(t) = 2.5t and
the area of a circle A = pr^2, graph these two equations and make a
conjecture about your answer before algebraically finding the answer.