Question

1. graph the function: (f(x)=\frac{x^{2}-4}{x^{2}-9}). state the intervals on which the function is increasing and decreasing.

Explanation:

Step1: Find derivative

Use quotient - rule. Let $u = x^{2}-4$, $v=x^{2}-9$. Then $u' = 2x$, $v'=2x$. The derivative $y'=\frac{u'v - uv'}{v^{2}}=\frac{2x(x^{2}-9)-2x(x^{2}-4)}{(x^{2}-9)^{2}}=\frac{-10x}{(x^{2}-9)^{2}}$.

Step2: Find critical points

Set $y' = 0$, so $-10x=0$, $x = 0$. Also, function is undefined at $x=\pm3$.

Step3: Test intervals

Test $(-\infty,-3)$: Let $x=-4$, $y'>0$, increasing. Test $(-3,0)$: Let $x=-1$, $y'>0$, increasing. Test $(0,3)$: Let $x = 1$, $y'<0$, decreasing. Test $(3,\infty)$: Let $x = 4$, $y'<0$, decreasing.

Answer:

Increasing on $(-\infty,-3)\cup(-3,0)$, decreasing on $(0,3)\cup(3,\infty)$