Question

g(x)= -4x^2 +36/ x+3

What key features does f(x), shown in the graph, share with g(x), shown in the equation? Select three options.

at least one x-intercept
at least one y-intercept
an oblique asymptote
a vertical asymptote
the domain of x

Answer 1

First simplify g(x):

g(x) = (-4x^2 + 36)/(x + 3) = -4(x^2 − 9)/(x + 3) = -4(x − 3)(x + 3)/(x + 3) = -4(x − 3) for x ≠ −3.

So g is the line y = −4x + 12 with a removable hole at x = −3. Therefore the graph (f) that matches g will have:

- at least one x-intercept (x = 3),
- at least one y-intercept (y = 12 at x = 0),
- the domain of x: all real numbers except x = −3.

It does NOT have a vertical asymptote (the factor cancels, so only a hole) and it does not have an oblique asymptote (the function is just a line except for the hole).