Question

Adding two functions results in h(x) = 4x, while multiplying the same functions results in j(x) = -5x2 - 12x - 4.

Which statements describe f(x) and g(x), the original functions? Select two options.
Both functions must be quadratic.
Both functions must have a constant rate of change.
Both functions must have a y-intercept of 0.
The rate of change of f(x) and g(x) must be opposites.
The y-intercepts of f(x) and g(x) must be opposites.

Answer 1

Answer: b and e

Step-by-step explanation:edge

Answer 2

When we multiply two functions, the degree is the sum of the original degrees. So, since the degree of the product is 2, we have two cases:

• One of the function has already degree 2, and the other is constant (degree 0)
• Both functions are linear.

The first case is actually impossible, because otherwise the sum would have degree 2 as well. So, we know that both
`f(x)` and
`g(x)` are linear. In other words, we have

`f(x)=ax+b,\quad g(x)=cx+d`

for some
`a,b,c,d \in \mathbb{R}`

We know that the sum is

`h(x)=4x=f(x)+g(x)=(a+c)x+(b+d)`

we deduce that

`a+c=4,\quad b+d=0`

So, we know that:

• Both functions must be quadratic. FALSE, otherwise the product would have degree 4;
• Both functions must have a constant rate of change. TRUE, linear functions have a constant rate of change;
• Both functions must have a y-intercept of 0. FALSE, it is only required that the sum of the y-intercepts is 0, they don't have to be both zero;
• The rate of change of f(x) and g(x) must be opposites. FALSE, their sum must be 4;
• The y-intercepts of f(x) and g(x) must be opposites. TRUE, their sum must be zero.