Increasing function

Definition

An increasing function is a function where, for any two points $x_1$ and $x_2$ in its domain, if $x_1 < x_2$, then $f(x_1) \leq f(x_2)$. This implies that the function values either rise or stay constant as one moves from left to right on the graph.

5 Must Know Facts For Your Next Test

1. A strictly increasing function satisfies $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.
2. Linear functions with positive slopes are examples of increasing functions.
3. The derivative of an increasing function is non-negative over its domain.
4. An increasing function does not necessarily have to be continuous; it can have jumps or discontinuities.
5. If the first derivative of a function is positive over an interval, the function is increasing on that interval.

Review Questions

• What condition must be satisfied for a function to be considered strictly increasing?
• How can you determine if a linear function is increasing by looking at its equation?
• What does it mean for the derivative of a function to be non-negative?

Related terms

Decreasing Function: A function where, for any two points $x_1$ and $x_2$ in its domain, if $x_1 < x_2$, then $f(x_1) \geq f(x_2)$. The values decrease or stay constant as one moves from left to right on the graph.

Derivative: A measure of how a function changes as its input changes. It represents the slope of the tangent line at any point on the graph of the function.

Monotonic Function: A function that is either entirely non-increasing or non-decreasing throughout its domain.