How do you know if a Function is Increasing or Decreasing?

How do you know if a Function is Increasing or Decreasing?

Increasing and decreasing functions

Definition:

(1) A function f is said to be an increasing function in ]a,b[, if x₁ < x₂ ⇒ f(x₁) < f(x₂) for all x₁, x₂ ∈ ]a,b[.
(2) A function f is said to be a decreasing function in ]a,b[, if x₁ < x₂ ⇒ f(x₁) < f(x₂), ∀ x₁, x₂ ∈ ]a,b[.
f(x) is known as non-decreasing if f’(x) ≥ 0 and non-increasing if f’(x) ≤ 0.

Monotonic function: A function f is said to be monotonic in an interval if it is either increasing or decreasing in that interval.
We summarize the results in the table below :

• Blank space indicates that the function may have any value at a₁.
• Question mark indicates that the behaviour of f cannot be inferred from the data.

Properties of monotonic functions

1. If f(x) is a strictly increasing function on an interval [a, b], then f⁻¹ exists and it is also a strictly increasing function.
2. If f(x) is a strictly increasing function on an interval [a, b] such that it is continuous, then f⁻¹ 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 ∈ (a, b), then f(x) is monotonically (strictly) increasing function on [a, b].
4. If f(x) is continuous on  such that f’(c) ≤ 0 [f’(c) > 0] for each c ∈ (a, b), then f(x) is monotonically (strictly) decreasing function on [a, b].
5. If f(x) and g(x) are monotonically (or strictly) increasing (or decreasing) functions on [a, b], then  is a monotonically (or strictly) increasing function on [a, b].
6. If one of the two functions f(x), g(x) is strictly (or monotonically) increasing and other a strictly (monotonically) decreasing, then gof(x) is strictly (monotonically) decreasing on [a, b].