Chapter 4.3, Problem 24AYU

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

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

To determine

To find: Ananlyze the given function.

Answer to Problem 24AYU

Explanation of Solution

Given:

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

Step 1: Factor $H$ to get

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

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

That is $x = -1, x = 1$

The domain $\left\{x\text{|}x \ne -1, x \ne 1\right\}$

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 $x\text{-intercept}$.

$y\text{-intercept}$ $-4$.

$x\text{-intercept}$ is 0.

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

$x = -1, x = 1$

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

Here horizontal asymptote $y = x^2$, not intersected.

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

Step 7: