Questions: Find f(a), f(a+h), and the difference quotient (f(a+h)-f(a))/h, where h ≠ 0. f(x)=4/(x+2) f(a)= f(a+h)= (f(a+h)-f(a))/h=

Find f(a), f(a+h), and the difference quotient (f(a+h)-f(a))/h, where h ≠ 0. f(x)=4/(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 $h \neq 0$. \[ \begin{array}{l} f(x)=\frac{4}{x+2} \\ f(a)= \end{array} \] $\square$ \[ f(a+h)= \] $\square$ \[ \frac{f(a+h)-f(a)}{h}= \] $\square$
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Solution

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Solution Steps

To solve this problem, we need to evaluate the function \( f(x) = \frac{4}{x+2} \) at \( x = a \) and \( x = a + h \). Then, we will compute the difference quotient \(\frac{f(a+h) - f(a)}{h}\).

  1. Evaluate \( f(a) \) by substituting \( a \) into the function.
  2. Evaluate \( f(a+h) \) by substituting \( a+h \) into the function.
  3. Compute the difference quotient \(\frac{f(a+h) - f(a)}{h}\).
Step 1: Evaluate \( f(a) \)

Given \( f(x) = \frac{4}{x+2} \), we substitute \( a = 3 \): \[ f(a) = \frac{4}{3+2} = \frac{4}{5} = 0.8 \]

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

Next, we substitute \( a = 3 \) and \( h = 0.1 \) into the function: \[ f(a+h) = f(3+0.1) = f(3.1) = \frac{4}{3.1+2} = \frac{4}{5.1} \approx 0.7843 \]

Step 3: Compute the Difference Quotient

The difference quotient is given by: \[ \frac{f(a+h) - f(a)}{h} = \frac{0.7843 - 0.8}{0.1} = \frac{-0.0157}{0.1} = -0.1569 \]

Final Answer

\[ \boxed{f(a) = 0.8} \] \[ \boxed{f(a+h) \approx 0.7843} \] \[ \boxed{\frac{f(a+h) - f(a)}{h} \approx -0.1569} \]

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