Monotonic function facts for kids

A monotonic function is a special kind of function in algebra and calculus. Think of a function as a rule that takes an input number and gives you an output number. A monotonic function is one where its "path" always goes in the same direction. It either always goes up, or it always goes down.

This means that the function's gradient (which is like its slope or steepness) never changes its sign. It won't go from positive to negative, or from negative to positive. For example, the function sin x is not monotonic because it goes up and down many times.

What is a Monotonic Function?

A function is like a machine that takes a number and gives you another number. For example, if you have the function f(x) = x + 2, when you put in 3, you get 5. When you put in 4, you get 6.

Understanding the Gradient

The gradient of a function tells you how steep its line or curve is, and in which direction it's going.

• If the gradient is positive, the function is going upwards (like walking up a hill).
• If the gradient is negative, the function is going downwards (like walking down a hill).
• If the gradient is zero, the function is flat for a moment (like standing on a flat part of the hill).

A monotonic function's gradient will always stay positive (or zero) or always stay negative (or zero). It never switches from going up to going down, or vice versa.

Always Increasing Functions

A function is called always increasing if, as you look at its graph from left to right, the line or curve always goes up or stays flat. It never goes down.

• This means that if you pick any two input numbers, and the first one is smaller than the second, then the output for the first number will also be smaller than or equal to the output for the second number.
• The gradient of an always increasing function is always positive or zero.

Always Decreasing Functions

A function is called always decreasing if, as you look at its graph from left to right, the line or curve always goes down or stays flat. It never goes up.

• This means that if you pick any two input numbers, and the first one is smaller than the second, then the output for the first number will be larger than or equal to the output for the second number.
• The gradient of an always decreasing function is always negative or zero.

What About Turning Points?

A turning point is a spot on a function's graph where it changes direction. For example, it might go from increasing to decreasing, or from decreasing to increasing.

• Since monotonic functions always go in one direction (either always up or always down), they do not have any turning points.
• However, a monotonic function can have a stationary point. This is a point where the gradient is momentarily zero, meaning the function is flat for an instant, but it continues in the same general direction afterward. Imagine a ramp that flattens out for a bit before continuing to go up.

The Derivative and Monotonic Functions

In calculus, the derivative of a function helps us find its gradient at any point.

• For a monotonic function, its derivative will never change its sign.
• If the function is always increasing, its derivative will always be positive or zero.
• If the function is always decreasing, its derivative will always be negative or zero.

See also

In Spanish: Función monótona para niños