Increasing and Decreasing Functions

Type	Condition on $I$	Graph
Increasing		Rising left to right
Decreasing		Falling left to right
Constant	$f(x_1)=f(x_2)$ for all $x_1,x_2$	Horizontal

Increasing Functions

A function is increasing if, as we move along the curve from left to right, the curve is rising. Formally, $f$ is increasing on an interval $I$ if

Decreasing Functions

A function is decreasing if, as we move along the curve from left to right, the curve is falling. Formally, $f$ is decreasing on an interval $I$ if

Constant Functions

A function is constant if its graph is horizontal. Formally, $f$ is constant on $I$ if

$$f(x_1)=f(x_2)\text{for all }{x}_1,x_2\text{ in }I.$$

Frequently Asked Questions

How does calculus determine where a function is increasing or decreasing?

If $f$ is differentiable on an interval $I$:

• f'(x) > 0 for all $x\in I$ implies $f$ is increasing on $I$.
• f'(x) < 0 for all $x\in I$ implies $f$ is decreasing on $I$.
• f'(x) = 0 for all $x\in I$ implies $f$ is constant on $I$.

This is one of the most useful results of differential calculus and underpins the first derivative test for local maxima and minima.

Can a function be both increasing and decreasing on overlapping intervals?

A function can be increasing on one interval and decreasing on another, but it cannot be both increasing and decreasing on the same interval (unless it is constant, which is the edge case). For example, $f(x)=x^2$ is decreasing on $(-\mathrm{\infty },0)$ and increasing on $(0,+\mathrm{\infty })$.

What is a monotone function?

A function that is increasing on its entire domain is called **monotonically increasing** (or strictly increasing). A function that is decreasing on its entire domain is called **monotonically decreasing**. Monotone functions are always one-to-one, because distinct inputs always produce distinct outputs.