Question

Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.

Answer 1

f(x) =`("x - 2")/("x + 1")`, x ≠ 0

For function to be increasing, f '(x) > 0

Then f '(x) = `(("x + 1") "d"/"dx" ("x - 2") - ("x - 2") "d"/"dx" ("x" + 1))/("x + 1")^2`

`= (("x + 1") - ("x - 2"))/("x + 1")^2 = ("x" + 1 - "x" + 2)/("x" + 1)^2`

`= 3/("x + 1")^2` > 0       ....[∵ (x + 1) ≠ 0, (x + 1)² > 0]

Thus, f(x) is an increasing function for x ≠ - 1.