Question

Evaluate the difference quotient for the given function. Simplify your answer. f(x) = 1 x , f(x) − f(a) x − a

Answer 1

Answer:

$\frac{f(x) - f(a)}{x - a}=\frac{-1}{xa}$

Step-by-step explanation:

Given

$f(x) = \frac{1}{x}$

Required

Evaluate $\frac{f(x) - f(a)}{x - a}$

We have:

$f(x) = \frac{1}{x}$

Next, we calculate f(a)

Substitute a for x in $f(x) = \frac{1}{x}$

$f(a) = \frac{1}{a}$

Substitute values for f(x) and f(a) in $\frac{f(x) - f(a)}{x - a}$

$\frac{\frac{1}{x} - \frac{1}{a}}{x - a}$

Take LCM of the numerator

$\frac{\frac{a - x}{xa} }{x - a}$

Split:

$\frac{a - x}{xa} /{x - a}$

Convert / to *

$\frac{a - x}{xa} *\frac{1}{x - a}$

Express a - x as -(x-a)

$\frac{-(x - a)}{xa} *\frac{1}{x - a}$

Divide numerator and denominator by x - a

$\frac{-1}{xa} *\frac{1}{1}$

$\frac{-1}{xa}$

Hence:

$\frac{f(x) - f(a)}{x - a}=\frac{-1}{xa}$