Question

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

Answer 1

`(f(x) - f(a))/(x - a)=(-1)/(xa)`

Explanation:

Given

`f(x) = (1)/(x)`

Required

Evaluate
`(f(x) - f(a))/(x - a)`

We have:

`f(x) = (1)/(x)`

Next, we calculate f(a)

Substitute a for x in
`f(x) = (1)/(x)`

`f(a) = (1)/(a)`

Substitute values for f(x) and f(a) in
`(f(x) - f(a))/(x - a)`

`((1)/(x) - (1)/(a))/(x - a)`

Take LCM of the numerator

`((a - x)/(xa) )/(x - a)`

Split:

`(a - x)/(xa) /{x - a}`

Convert / to *

`(a - x)/(xa) *(1)/(x - a)`

Express a - x as -(x-a)

`(-(x - a))/(xa) *(1)/(x - a)`

Divide numerator and denominator by x - a

`(-1)/(xa) *(1)/(1)`

`(-1)/(xa)`

Hence:

`(f(x) - f(a))/(x - a)=(-1)/(xa)`