Assignment 1: Operations With Functions

By Krista Floer

The problem is 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.

First, I came up with 3 different pairs of f(x) and g(x):

1. f(x) = 5x – 3 g(x) = 5x + 3

2. f(x) = -7x + 4 g(x) = -3x – 3

3. f(x) = 3x + 2 g(x) = -2x + 4

Note: For each transformation of the two linear equations, I have graphed the simplified version of each h(x). The first pair that gives h(x), it will be purple, the second is red and the last is blue. Plug-in is also needed to view the movies on this page.

i. Let’s look at adding 2 linear equations. When adding all of the pairs, we can see that h(x) in each case will still be a linear equation.

ii. Looking at multiplying f(x) and g(x), we can see that h(x) will now look like a parabola. We can see that the parabola is doing what one would expect; when the term is positive, the parabola opens upward and when the term is negative, the parabola opens downward.

iii. When dividing f(x) and g(x), the result is a hyperbola.

We can see that bottom portion of the purple graph goes over the red graph instead of being completely under the red graph. Seeing this I tried to figure out why this happened. I thought the reason this happened is because the two equations f(x) and g(x) are conjugates of each other. I decided to play with the constant in the numerator to see if this had an effect on the graph. From the movie below we can see that it did. I graphed , where n ranges from -10 to -1.

At one point, I found that the hyperbola would reflect and change direction. If the player is moved slow enough, you can see that as n approaches 3, the equation becomes y = 1. This is the exact point when the hyperbola changes directions. This is as n goes from -5 to 5.

iv. When composing f(x) and g(x), h(x) is again linear.

We know it will be linear because when we see an x in the first equation, we substitute in a linear equation. There will never be a case where these types of functions will construct anything other than a line given our constraints.