Question

for the function defined as follows, find a. f(x + h), b. f(x + h) - f(x), and c. f(x + h) - f(x)/h. f(x) = \frac{2}{x} a. f(x + h) = \frac{2}{x + h} b. f(x + h) - f(x) = \square

Explanation:

Step1: Substitute f(x+h) and f(x)

We know \( f(x + h)=\frac{2}{x + h} \) and \( f(x)=\frac{2}{x} \). So \( f(x + h)-f(x)=\frac{2}{x + h}-\frac{2}{x} \).

Step2: Find a common denominator

The common denominator of \( x + h \) and \( x \) is \( x(x + h) \). Rewrite the fractions: \( \frac{2x}{x(x + h)}-\frac{2(x + h)}{x(x + h)} \).

Step3: Subtract the numerators

Subtract the numerators: \( \frac{2x-2(x + h)}{x(x + h)} \). Expand the numerator: \( 2x-2x - 2h=-2h \). So the expression becomes \( \frac{-2h}{x(x + h)} \).

Answer:

\( \frac{-2h}{x(x + h)} \)