Assignment 1: Graphical Implications of Operating on Functions

by Shawn Broderick

I have chosen to explore Question 2 of Assignment 1. Here, again, are the directions:

2. Make up linear functions f (x) and g(x). Explore, with different pairs of f (x) and g(x) the graphs for
i. h(x) = f (x) + g(x)
ii. h(x) = f (x)g(x)
iii. h(x) = f (x)/g(x)
iv. h(x) = f (g(x))
Summarize and illustrate.

For this write up I am taking screen captures from Graphing Calculator, Version 3.5.

For this problem, we begin with linear functions f (x) and g(x). We define f (x) = 3x - 1 and g(x) = -2x + 3
Here are their corresponding graphs:

 f (x) = 3x - 1 g(x) = -2x + 3

i. Our first operation that we will investigate on these functions is addition:

Here is the graph of h(x) = f (x) + g(x) = (3x - 1) + (-2x + 3):

When h(x) is in its simplist form, we can see that it is h(x) = x + 2

ii. Our second operation that we will investigate on these functions is multiplication:

Here is the graph of h(x) = f (x)g(x) = (3x - 1)(-2x + 3):

Notice that it is a parabola. We can see that this parabola has some interesting characteristics. It opens down, the vertex is around (0.75, 2.1), the y-intercept is at -3, the x-intercepts are around 0.3 and 1.5.

iii. Our third operation that we will investigate on these functions is division:

Here is the graph of h(x) = f (x)/g(x) = (3x - 1)/(-2x + 3):

This graph is a rational function. It has an x-intercept at about 0.3 and a y-intercept at about -0.3. It also has an asymptote at x = 1.5 and a horizontal asymptote at y = 1.5.

iv. Our fourth operation that we will investigate on these functions is composition:

Here is the graph of h(x) = f (g(x)) = 3(-2x + 3) - 1:

Here we can see that the composition is another line. It has a y-intercept of 8 and a slope of about -6.