Answer:
B and D.
Step-by-step explanation:
We know that when we added the two functions together, we get:
However, when we multiplied them together, we get:
Let's first consider what we can determine from this.
If we add them together, we get a linear equation. However, this doesn't mean that our two original functions are linear since if we have, say, -x^2 and x^2-x+9, they will cancel and form a linear equation if we add them together.
<em>However</em>, since we know that if we multiply the two functions together, we get a linear equation, this means that both our original functions must be linear.
<em>But</em>, if we multiplied two linear functions, then we should get a quadratic, since x times x will yield x².
Therefore, this means that one of our linear functions is a horizontal line with no x variable. This is the only way to have a linear equation when multiplied.
Therefore, we have determined that both of our original functions are linear functions, and one (only one of them) is a horizontal line.
Let's go through each of the answer choices.
A) Both functions must be quadratic.
This is false as we determined earlier. If this was true, then the resulting function should be a quartic and not a line. A is false.
B) Both functions must have a constant rate of change.
Remember that all linear equations have a constant rate of change.
Since we determined that both our original functions are linear equation, this means that both our functions will have a constant rate of change.
So, B is true.
C) Both functions must have a y-intercept of 0.
Remember that one of our functions is a horizontal line.
If the y-intercept was 0, then the equation of our horizontal line will be:
And we know that anything multiplied by 0 will give us 0. However, the product of our function is -9x.
So, C cannot be true.
Rather, only our linear equation (not the horizontal line) may have a y-intercept of 0.
D) The rate of change of either f(x) or g(x) must be 0.
Remember that we determined that one of our lines must a horizontal.
Remember that horizontal lines have a slope of 0. In other words, the rate of change is 0.
So, D is true.
E) The y-intercepts of f(x) and g(x) must be opposites.
Well, since B and D is are true, this must be false since we can only select two options ;D
But, we can think about this. Note that if we multiply the two functions, we have a function <em>without</em> a y-intercept.
Remember that our horizontal line is <em>not</em> 0. So, the y-intercept of the horizontal line is a number.
So, the opposite of a number is another number.
So, if we multiply two non-zero numbers, we <em>must</em> get another number.
However, from our product, j(x)=-9x, we don't have another number. The y-intercept from this is 0.
Therefore, the two y-intercepts <em>cannot</em> be opposites of each other. If it was so, then we should have a y-intercept. So, E must be false.
In fact, this means that the y-intercept of our line (not the horizontal one) <em>must </em>be 0.
So, our answers are B and D.
And we're done!
Edit: Some (minor) errors in reasoning. Sorry!