Combining Functions

[CanadianEh]

1. The problem statement, all variables and given/known data

I was giving a graph of a combined function and four different equations and I need to select 2 of the equations and whether the graph shown is of addition, subtraction, multiplication or division. F(X)= X and H(X) = COS X and the combined graph is F(X)*H(X).

I need to justify my answer using key features of the graph as related to the initial functions

My attempt:

I could easily tell in my head that the graph shown on my paper was of X*cosX, but I'm just not sure how to explain it using the key features (domain, range, x and y ints, etc.). Essentially, I have to explain why I chose those two functions and why I chose multiplication.

- The new function is oscillating, which is why cos X is one of the functions.
- There are no asymptotes

What else can I say? Thanks in advance.

[Mark44]

Here are some ideas. You didn't say what the other two functions are, which makes it difficult to say why you chose f(x) = x and h(x) = cos x as the two you chose. As far as the arithmetic operation, think about how the graphs of f + h, f - h, fh, and f/h would look, based on the two functions you showed, and the other two that you didn't show.

For f + h you would have a graph that has the same oscillation as cos x, but with a central axis (for lack of a better word) that angles up to the right.

For f - h, which is equal to f + (-h), the graph would be similar to that of f + h, except that instead of adding cos x you would be adding its reflection across the x-axis.

For fh, when x > 0, you are multiplying by increasingly larger values, so you get the oscillation of the cosine factor, but the magnitude of the oscillation increases. When x < 0, this time you are multiplying by numbers that are getting more negative, which has a similar effect on the magnitude of oscillation, but also flips the cosine graph across the x-axis.

For f/h, because you are dividing by something that is periodically zero, there are going to be vertical asymptotes at each odd multiple of pi/2 (i.e., at +/-pi/2, +/-3pi/2, +/-5pi/2, and so on). At all other points you are going to be dividing by numbers in the interval [-1, 1], so there will be points on the graph that are identical to those on the graph of f(x) = x, and some that are identical to those on the graph of y = -x.

That should give you something to think about.