Let f: R → R and let f(x) > 0 for all .x ∈ R. Show that f(x) is strictly increasing if and only if the function g(x) = 1 /f(x) is strictly decreasing.

Step 1:

In this problem we have to show that f(x) is strictly increasing if and only if the function g(x) = 1/f(x) is strictly decreasing .

Step 2: definition of strictly increasing and strictly decreasing .

Strictly Increasing : A function is called strictly increasing , if for all x and y such that x < y then it has f(x) < f(y) . this is also called strictly monotonic increasing function.

Strictly Decreasing : A function is called strictly decreasing , if for all x and y such that x > y then it has f(x) > f(y) . this is also called strictly monotonic decreasing function.