Question

In Problems $7-44$, follow Steps $1$through $7$on page $228$to analyze the graph of each function.

$R\left(x\right)=\frac{x+1}{x\left(x+4\right)}$

Short answer

The graph for the function $R\left(x\right)=\frac{x+1}{x\left(x+4\right)}$is,

Step-by-step solution

Step 1 Given function is,

$R\left(x\right)=\frac{x+1}{x\left(x+4\right)}$

The domain of $R$is, $\left\{x|x\ne 0,-4\right\}$.

As the value $0$does not in the domain, there is no $y$-intercept.

The function $R$is in the lowest terms.

The real zero of the numerator satisfies the equation$x+1=0$. So, the$x$- intercept is,$-1$.

Step 2 The real zeros of the denominator are the real solutions of the equation.

$$\begin{gathered} x\left(x+4\right)=0 \\ x=0,-4 \end{gathered}$$

The lines $x=0,x=-4$are the vertical asymptotes of the graph.

In the given function, the degree of the numerator is less than the degree of the denominator. So, $R$is proper.

So, the line $y=0$is the horizontal asymptote of the graph.

Substitute $R\left(x\right)=0$in the given equation.

$$\begin{gathered} \frac{x+1}{x(x+4)}=0 \\ x+1=0 \\ x=-1 \end{gathered}$$

So, the graph intersects the horizontal asymptote at $\left(-1,0\right)$.

Step 3 Now plot a graph by using the known values.