Question

In Problems 25–30, use the graph shown to find

(a) The domain and range of each function

(b) The intercepts, if any

(c) Horizontal asymptote, if one exists

(d) Vertical asymptotes, if any

(e) Oblique asymptote, if one exists

Short answer

(a) The domain of the function is $\{x\overline{)x\in \mathrm{ℝ},x\ne 0\}}$and the range is the set of real numbers greater than $2$and less than $-2$

(b) There is no intercept for the function.

(c) There is no horizontal asymptote.

(d) The vertical asymptote is$y-axisorx=0$

(e) The oblique asymptote is$y=-x$

Step-by-step solution

Part (a) Step 1. Given

The graph:

Part (a) Step 2. Find the domain of the function.

From the graph, we can see that the graph approaches $0$, the function tends to $-\infty or\infty$.

So the value of the function is undefined at $0$.

The domain of the function is$\{x\overline{)x\in \mathrm{ℝ},x\ne 0\}}$

Part (a) Step 3. Find the range

From the graph, if the domain approaches $-\infty or\infty$, the graph approaches the line .

So the range of the function is set of real numbers greater than $2$and less than $-2$

Part (b) Step 1. Intercept of the function

The intercept is the point at which the function touches the respective axis.

Part (b) Step 2. Intercept of the function

There is no intercept for the given function since the line does not touch the coordinate axis.

Part (c) Step 1. Horizontal asymptote

If $x\to -\infty ,\infty$, the values of the function approach some fixed number $L$, then the line $y=L$is the horizontal asymptote of the graph.

Part (c) Step 2. Find the horizontal asymptote

The graph of the rational function cannot have both horizontal and oblique asymptote.

Since the graph have oblique asymptote, it does not have an horizontal asymptote.

Part (d) Step 1. Vertical asymptote

If $x$approaches some number $c$, the value of $\left|R\left(x\right)\right|\to \infty$, then the line $x=c$is the vertical asymptote of the graph.

Part (d) Step 2. Find the vertical asymptote.

The graph approaches $\infty$at the constant $x=0$

So the vertical asymptote is $y-axisorx=0$

Part (e) Step 1. Oblique asymptote

If $f\left(x\right)=ax+b,a\ne 0$then the line $y=ax+b$is an oblique asymptote of the graph of $R$.

Part (e) Step 2. Find the oblique asymptote

From the graph, we can see that the function approaches the point $(0,0),(-1,1)asx\to \pm \infty$

$$\begin{gathered} m=\frac{1-0}{-1-0} \\ =-1 \end{gathered}$$

So the oblique asymptote of the line is given by$y=-x$