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(cx + d)                                           g(f(x)) = g(ax + b)

= a(cx + d) + b                                               = c(ax + b) + d

= acx + (ad + b)                                             = cax + (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:

f(g(x)) = f() = 2() + 1 =

g(f(x)) = g(2x + 1) =  =

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.

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.