Questions: Matching graphs with rational functions: Two vertical asymptotes Choose the graph of each function from the choices below. (a) Which is the graph of f(x) = 8 / (x^2 - 2x - 8) ? (choose one) (b) Which is the graph of g(x) = (3x + 6) / (x^2 - 16) ? (choose one)

Solution Steps

To match the graphs with the given rational functions, we need to identify the vertical asymptotes of each function. Vertical asymptotes occur where the denominator of the function is zero, provided the numerator is not zero at those points.

(a) For the function \( f(x) = \frac{8}{x^2 - 2x - 8} \), we find the vertical asymptotes by setting the denominator equal to zero and solving for \( x \).

(b) For the function \( g(x) = \frac{3x + 6}{x^2 - 16} \), we similarly find the vertical asymptotes by setting the denominator equal to zero and solving for \( x \).

For the function \( f(x) = \frac{8}{x^2 - 2x - 8} \), we set the denominator equal to zero to find the vertical asymptotes:

\[ x^2 - 2x - 8 = 0 \]

Factoring the quadratic, we find:

\[ (x + 2)(x - 4) = 0 \]

Thus, the vertical asymptotes are:

\[ x = -2 \quad \text{and} \quad x = 4 \]

For the function \( g(x) = \frac{3x + 6}{x^2 - 16} \), we also set the denominator equal to zero:

\[ x^2 - 16 = 0 \]

Factoring gives us:

\[ (x - 4)(x + 4) = 0 \]

Thus, the vertical asymptotes are:

\[ x = -4 \quad \text{and} \quad x = 4 \]

Final Answer