find any points of discontinuity for the function. f(x)=(x^2 - 4x - 32)/(x^2 - 15x + 56) hole: x = vertical…

Explanation:

Step1: Factor the numerator and denominator

Factor $x^{2}-4x - 32=(x - 8)(x+4)$ and $x^{2}-15x + 56=(x - 7)(x - 8)$. So $f(x)=\frac{(x - 8)(x + 4)}{(x - 7)(x - 8)}$.

Step2: Find the hole

A hole occurs when a factor can be canceled out. Since we can cancel out $(x - 8)$ (where $x\neq8$), the hole is at $x = 8$.

Step3: Find the vertical - asymptote

The vertical asymptote occurs when the denominator of the simplified function is zero. After canceling $(x - 8)$, the denominator of the simplified function $y=\frac{x + 4}{x - 7}$ is zero when $x=7$. So the vertical asymptote is $x = 7$.

Step4: Find the horizontal - asymptote

Since the degree of the numerator and the denominator of the simplified function $y=\frac{x + 4}{x - 7}$ are the same (both degree 1), the horizontal asymptote is $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$. The leading coefficient of the numerator is 1 and the leading coefficient of the denominator is 1, so $y = 1$.

Answer:

Hole: $x = 8$ Vertical Asymptote: $x = 7$ Horizontal Asymptote: $y = 1$