Questions: Find f(a), f(a+h), and the difference quotient (f(a+h)-f(a))/h, where h ≠ 0. f(x)=x/(x+2) f(a)= f(a+h)= (f(a+h)-f(a))/h=

Solution Steps

To solve this problem, we need to evaluate the function $f(x) = \frac{x}{x+2}$ at specific points and then compute the difference quotient. First, substitute $a$ into the function to find $f(a)$. Next, substitute $a+h$ into the function to find $f(a+h)$. Finally, use these results to calculate the difference quotient $\frac{f(a+h)-f(a)}{h}$.

To find $f(a)$, substitute $a$ into the function $f(x) = \frac{x}{x+2}$. This gives: $$f(a) = \frac{a}{a+2}$$

To find $f(a+h)$, substitute $a+h$ into the function: $$f(a+h) = \frac{a+h}{a+h+2}$$

The difference quotient is given by: $$\frac{f(a+h) - f(a)}{h} = \frac{\frac{a+h}{a+h+2} - \frac{a}{a+2}}{h}$$ Simplifying this expression, we find: $$\frac{f(a+h) - f(a)}{h} = \frac{2}{a^2 + ah + 4a + 2h + 4}$$

Final Answer