Question

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

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

Short answer

The graph is

Step-by-step solution

Given information

Given function$f\left(x\right)=x+\frac{1}{x}$

Factoring the numerator and denominator

Factoring, we get

$$\begin{gathered} f\left(x\right)=x+\frac{1}{x} \\ f\left(x\right)=\frac{x^2+1}{x} \end{gathered}$$

Calculate the domain

Setting denominator to zero, we get

The domain is all real numbers except$0$

Calculate the x intercept

Setting numerator to zero, we get

$$\begin{gathered} x^2+1=0 \\ {x}^2=-1 \end{gathered}$$

There is no role="math" localid="1646211568725" $x,y$intercept as$f\left(0\right)=undefined$

Calculate the vertical asymptote

The zeroes of denominator are $x=0$so the vertical asymptote is$x=0$

Calculate the horizontal asymptote

The degree of numerator is greater than denominator so there is an oblique asymptote

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

So $y=x$is the oblique asymptote

$$\begin{gathered} f\left(x\right)=\frac{x^2+1}{x} \\ =x \\ {x}^2+1=x^2 \\ 0=-1 \end{gathered}$$

It doesn't intersect$f$

Sketching the graph

The graph is