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=

Transcript text: Find $f(a), f(a+h)$, and the difference quotient $\frac{f(a+h)-f(a)}{h}$, where $h \neq 0$. \[ \begin{array}{c} f(x)=\frac{x}{x+2} \\ f(a)=\square \\ f(a+h)=\square \\ \frac{f(a+h)-f(a)}{h}=\square \end{array} \] Need Help? Read It Watch it

Solution

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 $\frac{f(a+h)-f(a)}{h}$.

Step 1: Evaluate $ f(a) $

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

Step 2: Evaluate $ f(a+h) $

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

Step 3: Calculate the Difference Quotient

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: \[ \frac{f(a+h) - f(a)}{h} = \frac{2}{a^2 + ah + 4a + 2h + 4} \]