Properties of Monotonic Functions
Recall: A function $f(x)$ is increasing on an interval if $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$, and decreasing if $x_1 < x_2 \Rightarrow f(x_1) > f(x_2)$. A function that is either entirely increasing or entirely decreasing on an interval is called monotonic on that interval.
Now that we understand what monotonic functions are, let's explore five powerful properties that allow us to analyze complex functions, solve equations, and prove uniqueness of solutions.
Property 1: Uniqueness (The Horizontal Line Test)
Statement: A monotonic function takes each value at only one argument value. That is, if $f$ is monotonic on $X$ and $f(x_1) = f(x_2)$, then $x_1 = x_2$.
Consequences:
- Any horizontal line $y = a$ intersects the graph of a monotonic function at most once.
- The equation $f(x) = a$ has at most one root if $f$ is monotonic.
Why is this true?
Suppose $f$ is increasing and $f(x_1) = f(x_2)$. If $x_1 < x_2$, then $f(x_1) < f(x_2)$ — contradiction. If $x_1 > x_2$, then $f(x_1) > f(x_2)$ — contradiction. So $x_1 = x_2$. ✓
Interactive visualization — available in the full lesson