Decreasing function

Definition

A decreasing function is one where, for any two points $x_1$ and $x_2$ in its domain, if $x_1 < x_2$, then $f(x_1) \geq f(x_2)$. This indicates that as the input increases, the output either stays the same or decreases.

5 Must Know Facts For Your Next Test

1. A function is strictly decreasing if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.
2. Graphically, a decreasing function has a downward slope from left to right.
3. In calculus terms, a function is decreasing where its derivative is less than or equal to zero.
4. For linear functions of the form $f(x) = mx + b$, the function is decreasing if and only if the slope $m$ is negative.
5. Decreasing functions are important in identifying intervals of monotonicity in more complex functions.

Review Questions

• How can you determine if a linear function is decreasing by looking at its equation?
• What is the graphical representation of a decreasing function?
• Explain how the derivative of a function relates to whether it is increasing or decreasing.

Related terms

Increasing Function: A function where, for any two points $x_1$ and $x_2$ in its domain, if $x_1 < x_2$, then $f(x_1) \leq f(x_2)$.

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

Derivative: The instantaneous rate of change of a function with respect to one of its variables. Used to determine increasing and decreasing behavior.