Question

consider the following rational functions.
$f(x)=\frac{x^{2}+1}{2x^{2}-32}$
$g(x)=\frac{2x+2}{x^{2}+2x-3}$
choose the graph of each function from the choices below.

Explanation:

Step1: Find asymptotes of $f(x)$

Vertical Asymptotes:

Set denominator to 0:
$2x^2 - 32 = 0$
$2(x^2 - 16) = 0$
$x^2 - 16 = 0$
$(x-4)(x+4)=0$
$x=4, x=-4$

Horizontal Asymptote:

Degrees of numerator/denominator equal, take ratio of leading coefficients:
$\lim_{x\to\pm\infty} f(x) = \frac{1}{2}$

Check sign at $x=0$:

$f(0)=\frac{0^2+1}{2(0)^2-32}=-\frac{1}{32}$, so $f(x)$ is negative at $x=0$, matching Graph A's lower curve crossing the y-axis below 0.

Step2: Find asymptotes of $g(x)$

Vertical Asymptotes:

Set denominator to 0:
$x^2 + 2x - 3 = 0$
$(x+3)(x-1)=0$
$x=-3, x=1$

Horizontal Asymptote:

Degree of denominator > numerator, so:
$\lim_{x\to\pm\infty} g(x) = 0$

Check sign at $x=0$:

$g(0)=\frac{2(0)+2}{0^2+2(0)-3}=-\frac{2}{3}$, so $g(x)$ is negative at $x=0$, matching Graph C's lower curve crossing the y-axis below 0.

Answer:

• $f(x) = \frac{x^2 + 1}{2x^2 - 32}$ corresponds to Graph A
• $g(x) = \frac{2x+2}{x^2 + 2x - 3}$ corresponds to Graph C