Question

True or False? Justify your answers.

(1) If a function defined on R is both increasing and decreasing, then it is constant.
(2) The function f: R\{8} → R defined by f(x)= (x-8)²/x-8 + 100 is surjective.

Answer 1

If a function is both increasing and decreasing, it must be constant. The function f(x) = (x-8)²/(x-8) + 100 is not surjective due to a discontinuity at x = 8.

Step-by-step explanation:

(1) The statement is true. If a function is both increasing and decreasing, it means that for any two different inputs x1 and x2, if x1 < x2 then f(x1) < f(x2) and if x1 > x2 then f(x1) > f(x2). But since the function is increasing and decreasing, there must exist some x and y such that x < y and x > y, which is not possible. Therefore, the function must be constant.

(2) The statement is false. To determine if a function is surjective, we need to check if every element in the codomain has a pre-image in the domain. In this case, the function f(x) = (x-8)²/(x-8) + 100 is not surjective because there is no pre-image for the value 100 in the codomain. The function has a discontinuity at x = 8 and is undefined at that point.