Answer
a) $f(0)=0$
$f(2)=\frac{\sqrt2}{3}$
$f(a+2)=\frac{\sqrt{a+2}}{a+3}$
b) Domain: [0, ∞)
c) Avg. rate of change: $\frac{\sqrt10}{88}-\frac{\sqrt2}{24}$
Work Step by Step
a) $f(0)=\frac{\sqrt0}{0+1}=\frac{0}{1}=0$
$f(2)=\frac{\sqrt2}{2+1}=\frac{\sqrt2}{3}$
$f(a+2)=\frac{\sqrt{a+2}}{a+2+1}=\frac{\sqrt{a+2}}{a+3}$
b) To find the domain, one must look at all the restrictions in the function. In this case, there are two of them. The first one is that the denominator cannot be zero, therefore $x\ne -1$. The second one is that the number inside the square root cannot be negative, thus $x\geq 0$.
Since the first restriction is part of the second, there is only one restriction. So, all $x$ values must be equal to or greater than zero. [0,∞)
c) Avg. rate of change:
$\frac{f(b)-f(a)}{b-a}$
$\frac{f(10)-f(2)}{10-2}$
$\frac{\frac{\sqrt10}{10+1}-\frac{\sqrt2}{2+1}}{8}$
$\left(\frac{\sqrt10}{11}-\frac{\sqrt2}{3}\right)\cdot \frac{1}{8}$ Apply the distributive property
$\frac{\sqrt10}{11\cdot 8}-\frac{\sqrt2}{3\cdot8}$
$\frac{\sqrt10}{88}-\frac{\sqrt2}{24}$
