# How to Find the Increasing or Decreasing Functions?

Increasing and decreasing functions are functions in calculus for which the value of $$f(x)$$ increases and decreases respectively with the increase in the value of $$x$$.

If the value of $$f(x)$$ increases with the increasing value of $$x$$, the function is said to be increasing, and if the value of $$f(x)$$ decreases with the increasing value of $$x$$, the function is decreasing.

## A step-by-step guide to increasing and decreasing functions

Increasing and decreasing functions are functions whose graphs go up and down respectively by moving to the right of the $$x$$-axis. Increasing and decreasing functions are also called non-decreasing and non-increasing functions.

### Increasing and decreasing functions definition

Increasing function: The function $$f(x)$$ in the interval $$I$$ is increasing on anif for any two numbers $$x$$ and $$y$$ in $$I$$ such that $$x<y$$, we have $$f(x) ≤ f(y)$$.

Decreasing function: The function $$f(x)$$ in the interval $$I$$ is decreasing if for any two numbers $$x$$ and $$y$$ in $$I$$ such that $$x<y$$, we have $$f(x) ≥ f(y)$$.

Strictly increasing function: A function $$f(x)$$ is called to be strictly increasing on an interval $$I$$ if for any two numbers $$x$$ and $$y$$ in $$I$$ such that $$x<y$$, we have $$f(x) < f(y)$$.

Strictly decreasing function: A function $$f(x)$$ is called to be strictly decreasing on an interval $$I$$ if for any two numbers $$x$$ and $$y$$ in $$I$$ such that $$x<y$$, we have $$f(x)>f(y)$$.

### Rules to check increasing and decreasing functions

We use a derivative of a function to check whether the function is increasing or decreasing. Suppose a function $$f(x)$$ is differentiable on an open interval $$I$$, then we have:

• If $$f'(x) ≥ 0$$ on $$I$$, the function is said to be an increasing function on $$I$$.
• If $$f'(x)≤ 0$$ on $$I$$, the function is said to be a decreasing function on $$I$$.

Note: The first derivative of a function is used to check for increasing and decreasing functions.

### Properties of increasing and decreasing functions

• If the functions $$f$$ and $$g$$ are increasing functions on an open interval $$I$$, then the sum of the functions $$f+g$$ is also increasing on this interval.
• If the functions $$f$$ and $$g$$ are decreasing functions on an open interval $$I$$, then the sum of the functions $$f+g$$ is also decreasing on this interval.
• If the function $$f$$ is an increasing function on an open interval $$I$$, then the opposite function $$-f$$ decreases on this interval.
• If the function $$f$$ is a decreasing function on an open interval $$I$$, then the opposite function $$-f$$ is increasing on this interval.
• If the function $$f$$ is an increasing function on an open interval $$I$$, then the inverse function $$\frac{1}{f}$$ is decreasing on this interval.
• If the function $$f$$ is a decreasing function on an open interval $$I$$, then the inverse function $$\frac{1}{f}$$ is increasing on this interval.
• If the functions $$f$$ and $$g$$ are increasing functions on an open interval $$I$$ and $$f, g ≥ 0$$ on $$I$$, then the product of the functions $$fg$$ is also increasing on this interval.
• If the functions $$f$$ and $$g$$ are decreasing functions on an open interval $$I$$ and $$f, g ≥ 0$$ on $$I$$, then the product of the functions $$fg$$ is also decreasing on this interval.

## Exercises for Increasing and Decreasing Functions

### Determine the intervals at which the function is increasing.

• $$\color{blue}{f\left(x\right)=x\:ln\:x}$$
• $$\color{blue}{f\left(x\right)=4x-x^2}$$

### Determine the intervals at which the function is decreasing.

• $$\color{blue}{f\left(x\right)=5-2x-x^2}$$
• $$\color{blue}{f\left(x\right)=xe^{3x}}$$
• $$\color{blue}{(\frac{1}{e}, \infty)}$$
• $$\color{blue}{(-\infty,2)}$$
• $$\color{blue}{(-1,\infty)}$$
• $$\color{blue}{\left(-\infty ,-\frac{1}{3}\right)}$$

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