Increasing function

Definition

An increasing function is a function where the value of the output increases as the input increases. Mathematically, for any two values $x_1$ and $x_2$ such that $x_1 < x_2$, the function $f(x)$ satisfies $f(x_1) \leq f(x_2)$.

5 Must Know Facts For Your Next Test

1. In an increasing function, if $x_1 < x_2$ then $f(x_1) \leq f(x_2)$ holds true.
2. If the inequality is strict, i.e., $f(x_1) < f(x_2)$ when $x_1 < x_2$, then the function is called strictly increasing.
3. The slope of the graph of an increasing linear function is always positive.
4. In calculus terms, if the derivative of a function $f'(x) > 0$ for all $x$ in its domain, then it is an increasing function.
5. An increasing function may have flat regions where the slope is zero but does not decrease at any point.

Review Questions

Related terms

Decreasing Function: A decreasing function is one where as the input increases, the output decreases. Mathematically, for any two values $x_1$ and $x_2$ such that $x_1 < x_2$, then $f(x_1) \geq f(x_2)$.

Linear Function:

A linear function has the form $f(x) = mx + b$, where each term represents constants or variables with only degree one.

Rate of Change:

The rate of change measures how much a quantity changes over a specific period or interval. In functions, it often refers to $\Delta y / \Delta x$.