Adding two functions results in h(x) = 4x, while multiplying the same functions results in j(x) = -5×2 – 12x – 4. Which statements des

Question

Adding two functions results in h(x) = 4x, while multiplying the same functions results in j(x) = -5×2 – 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.

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Autumn 2 months 2021-10-15T23:51:50+00:00 1 Answer 0 views 0

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    2021-10-15T23:53:04+00:00

    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.

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