2. Consider the following functions. m(x)=x^2-1 n(x)=sqrt (x) p(x)=(1)/(x-3) a. Determine (mcirc ncirc p)(12) b. Write the equation for p(m(x)) and determine its domain.

## Step 1: Calculate $p(12)$<br />### We are given $p(x) = \frac{1}{x-3}$. Substituting $x=12$, we get $p(12) = \frac{1}{12-3} = \frac{1}{9}$.<br /><br />## Step 2: Calculate $n(p(12))$<br />### We are given $n(x) = \sqrt{x}$. Substituting $x = p(12) = \frac{1}{9}$, we get $n(p(12)) = n(\frac{1}{9}) = \sqrt{\frac{1}{9}} = \frac{1}{3}$.<br /><br />## Step 3: Calculate $m(n(p(12)))$<br />### We are given $m(x) = x^2 - 1$. Substituting $x = n(p(12)) = \frac{1}{3}$, we get $m(n(p(12))) = m(\frac{1}{3}) = (\frac{1}{3})^2 - 1 = \frac{1}{9} - 1 = \frac{1-9}{9} = -\frac{8}{9}$.<br /><br />## Step 4: Determine $p(m(x))$<br />### We are given $m(x) = x^2 - 1$ and $p(x) = \frac{1}{x-3}$. Substituting $m(x)$ into $p(x)$, we get $p(m(x)) = p(x^2 - 1) = \frac{1}{(x^2 - 1) - 3} = \frac{1}{x^2 - 4}$.<br /><br />## Step 5: Determine the domain of $p(m(x))$<br />### The function $p(m(x)) = \frac{1}{x^2 - 4}$ is undefined when the denominator is zero. $x^2 - 4 = 0$ implies $x^2 = 4$, so $x = \pm 2$. Therefore, the domain of $p(m(x))$ is all real numbers except $x = 2$ and $x = -2$. In interval notation, the domain is $(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$.