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) = \frac{-2h}{x(x + h)} c. \frac{f(x + h) - f(x)}{h} = \square

Explanation:

Step1: Substitute the expression from part b

We know from part b that \( f(x + h) - f(x)=\frac{-2h}{x(x + h)} \). Now we need to divide this by \( h \). So we have \( \frac{\frac{-2h}{x(x + h)}}{h} \).

Step2: Simplify the division

Dividing a fraction by a number is the same as multiplying the fraction by the reciprocal of that number. So \( \frac{\frac{-2h}{x(x + h)}}{h}=\frac{-2h}{x(x + h)}\times\frac{1}{h} \). The \( h \) in the numerator and denominator will cancel out (assuming \( h
eq0 \)), leaving us with \( \frac{-2}{x(x + h)} \).

Answer:

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