Chapter 2.6, Problem 24E

Solution

The asymptotes and intercepts of function and graph the function.

The vertical asymptotes, $x=1$ , $x=-3$ , horizontal asymptotes, $y=0$ with intercepts $\left(-2,0\right)$ and $\left(0,-\frac{2}{3}\right)$ .

Given:

The function is $g\left(x\right)=\frac{x+2}{x^2+2x-3}$ .

Concept Used:

If a polynomial function in the form $y=\frac{f\left(x\right)}{g\left(x\right)}=\frac{\left(a_n{x}^n+\mathrm{....}\right)}{\left(b_m{x}^m+\mathrm{....}\right)}$ , then, vertical asymptotes is given by real zeros of denominator provided it should not be zero of numerator.

And,

The end behaviour asymptote given by $\underset{x\to {\infty }^{-}}{\lim }\frac{f\left(x\right)}{g\left(x\right)}\text{ or }\underset{x\to {\infty }^{+}}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is horizontal asymptotes.

The condition can be concluded as,

1) If $n<m$ , the end behaviour is horizontal asymptotes that is $y=0$ ,

2) If $n=m$ the end behaviour is horizontal asymptotes is horizontal asymptotes that is $y=\frac{a_n}{b_n}$

3) If $n>m$ ,the quotient $q\left(x\right)$ of $f\left(x\right)/g\left(x\right)$ is the end behaviour asymptotes, where $q\left(x\right)$ is given by $f\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$ with $r\left(x\right)$ is remainder of $f\left(x\right)/g\left(x\right)$ . Hence, no horizontal asymptotes.

And,

The x -intercept is given by zeros of numerator that are not zero of denominator. And y -intercept is given by $f\left(0\right)$ .

From using asymptotes and intercepts the graph can be drawn.

Calculation:

Consider the function, $g\left(x\right)=\frac{x+2}{x^2+2x-3}$ .

To find vertical asymptotes, find the zeros of denominator, so factorize the denominator $x^2+2x-3$ :

  $\begin{array}{c}{x}^2+2x-3=x^2+3x-x-3\\ =x\left(x+3\right)-1\left(x+3\right)\\ =\left(x+3\right)\left(x-1\right)\end{array}$

Thus, the zeros of denominator of $f\left(x\right)$ is $x=-3$ and $x=1$ , Means the vertical asymptotes are $x=-3$ and $x=1$ .

To find end behaviour asymptotes, find $m\text{ and }n$ by comparing, $g\left(x\right)=\frac{x+2}{x^2+2x-3}$ with $y=\frac{f\left(x\right)}{g\left(x\right)}=\frac{\left(a_n{x}^n+\mathrm{....}\right)}{\left(b_m{x}^m+\mathrm{....}\right)}$ .

Since, $n<m$ end behaviour asymptotes is horizontal asymptotes, $y=0$ .

Now find the intercept, the x -intercept is given by zeros of numerator that are not zero of denominator.

Thus, x -intercept is $x+2=0\text{ or }x=-2$ or $\left(-2,0\right)$ .

And y -intercept is given by $g\left(0\right)$ , so from $g\left(x\right)=\frac{x+2}{x^2+2x-3}$ .

  $\begin{array}{c}g\left(0\right)=\frac{\left(0\right)+2}{{\left(0\right)}^2+2\left(0\right)-3}\\ =\frac{2}{-3}\\ =-\frac{2}{3}\end{array}$

Thus y -intercept is $y=-\frac{2}{3}$ or $\left(0,-\frac{2}{3}\right)$ .

Hence, using vertical asymptotes, $x=1$ , $x=-3$ , horizontal asymptotes, $y=0$ with intercepts $\left(-2,0\right)$ and $\left(0,-\frac{2}{3}\right)$

the graph obtain is as:

  

Conclusion:

The vertical asymptotes, $x=1$ , $x=-3$ , horizontal asymptotes, $y=0$ with intercepts $\left(-2,0\right)$ and $\left(0,-\frac{2}{3}\right)$ .