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Increasing Function

Tool to calculate if a function is increasing / monotonic or on which interval is increasing or strictly increasing.

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Increasing Function -

Tag(s) : Functions

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# Increasing Function

## Increasing Function Calculator

### Increasing Interval Finder

 Monotony Strictly increasing Weakly increasing

### What is an increasing function? (Definition)

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 $] 0 ; +\infty [$

### How to determine if a function is increasing?

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.

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

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.

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