Chapter 4.5, Problem 29AYU

In Problems 7-50, follow Steps 1 through 7 on page 234 to analyze the graph of each function.

$H\left(x\right)=\frac{x^2-1}{x^4-16}$

To determine

To find: Ananlyze the given function.

Answer to Problem 29AYU

Explanation of Solution

Given:

$H\left(x\right) = \frac{x^2 - 1}{x^4 - 16}$

Step 1: Factor $\text{R}$ to get

$H\left(x\right) = \frac{x^2 - 1}{x^4 - 16}$

$H\left(x\right) = \frac{x^2 - 1}{{\left(x^4 + 1\right)}^2\left(x + 2\right)\left(x - 2\right)}$

That is $x = -2, x = 2$.

The domain $\left\{x\text{|}x \ne -2, x \ne 2\right\}$

Step 2: $H$ is in lowest terms

Step 3: Locate the intercepts of the graph. Use multiplicity to determine the bahaviour of the graph of $H$ at each $x\text{-intercept}$.

$y\text{-intercept}$ $\frac{1}{16}$

$x\text{-intercept}$ is $-1$ and 1.

Step 4: Locate the vertical asymptotes. Determine the behaviour of the graph on either side of each vertical asymptote.

$x = -2, x = 2$

Step 5: Locate the horizontal or oblique asymptote. Determine points, if any, at which the graph of $H$ intersects this asymptote.

Here horizontal asymptote $y = 0$, intersected at $\left(-1, 0\right)$ and $\left(1, 0\right)$.

Step 6: Graph $H$ using a graphing utility.

Step 7: