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 yintercept is at 3, the xintercepts 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 xintercept at about 0.3 and a yintercept 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 yintercept of 8 and a slope of about 6.