Increasing Function

Question 1

What is an increasing function? (Definition)

Answer

A function $ f $ is strictly increasing if for any $$ x_1 < x_2 , f(x_1) < f(x_2) $$

In other words, $ f $ has an increasing direction of variation, when $ x $ increases, $ f(x) $ also increases (not necessarily by the same quantity).

A function is said to be increasing (not strictly, in the broad sense) if for all $$ x_1 < x_2 , f(x_1) \leq f(x_2) $$

Example: The function $ f(x) = x + 1 $ is increasing over its whole domain of definition $ \mathbb{R} $, hence its monotony

The growth of a function can also be defined over an interval.

Example: The function $ f(x) = x^2 $ is strictly increasing over $ \mathbb{R}^+ $ also noted $ x > 0 $ or also $ ] 0 ; +\infty [ $

Question 2

How to determine if a function is increasing?

Answer

Several methods allow to know if a function is increasing (study of the direction of variation):

— From its derivative: if the derivative of the function is greater than $ 0 $ then the function is increasing.

Example: The derivative of the function $ f(x) = x^2+2 $ is $ f'(x) = 2x $, the calculation of the inequation $ f'(x) > 0 $ is solved $ x > 0 $ so the function $ f $ is increasing when $ x > 0 $

— From its equation: Some functions are notoriously increasing, ie. the exponential function, the logarithm function, odd degree monomers, etc.

Example: $ \exp(x) $ is increasing over $ \mathbb{R} $

— From the curve of the function: an increasing function has its curve which is directed upwards.

Question 3

How to determine if a linear/affine function is increasing?

Answer

A linear function of the form $ f(x) = ax + b $ is monotonic and strictly increasing over $ \mathbb{R} $ when the coefficient $ a $ is strictly positive ($ a > 0 $).

If $ a $ is negative then the function is decreasing.

If $ a = 0 $ then the function is constant.

Increasing Function