Question

Evaluate the difference quotient for the given function. Simplify your answer.

f(x) = x + 5/ x + 1 , f(x) − f(3)/x − 3

Answer 1

I suppose you mean

`f(x) = (x+5)/(x+1)`

Then

`f(3) = (3+5)/(3+1) = \frac84 = 2`

and the difference quotient is

`(f(x)-f(3))/(x-3) = ((x+5)/(x+1)-2)/(x-3) \\\\ (f(x)-f(3))/(x-3) = ((x+5-2(x+1))/(x+1))/(x-3) \\\\ (f(x)-f(3))/(x-3) = (-x+3)/((x+1)(x-3)) \\\\ (f(x)-f(3))/(x-3) = \boxed{-(x-3)/((x+1)(x-3))}`

If it's the case that x ≠ 3, then (x - 3)/(x - 3) reduces to 1, and you would be left with

`(f(x)-f(3))/(x-3)\bigg|_(x\\eq3) = -\frac1{x+1}`