Properties of Monotonic Functions

1. If f(x) is strictly increasing function on an interval [a,b], then f ˉ¹ exists and it is also a strictly increasing function.

2. If f(x) is strictly increasing function on an interval [a,b], such that it is continuous, then f ˉ¹ is continuous on [f(a),f(b)].

3. If f(x) is continuous on [a,b] such that f ‘ (c) ≥0 (f ‘ (c)>0) for each c Є(a,b), then f(x) is monotonically increasing function on [a,b].

4. If f(x) is continuous on [a,b] such that f ‘ (c) ≤0 (f ‘ (c)<0) for each c Є(a,b), then f(x) is monotonically decreasing function on [a,b].

5. If f(x) and g(x) are monotonically increasing (or decreasing) functions on [a,b], then

6. If one of the functions is monotonically increasing and the other is monotonically decreasing, then gof is a monotonically decreasing function on [a,b].