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

• What condition must hold for a function to be considered increasing?
• How does the slope of a linear increasing function compare to zero?
• Describe how you can determine if a non-linear function is increasing using calculus.

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 $\Delta y / \Delta x$.