Bounded and Unbounded Functions
Big Idea: When we study functions, one of the most important questions we can ask is: "Does this function's output stay within limits, or does it grow without bound?" This concept is called boundedness.
1. Bounded Below
Definition: A function $f$ is bounded below if there exists a number $a$ such that for every $x$ in the domain $D(f)$:
$$ f(x) \geq a $$
What does this mean geometrically?
- The graph of $f$ lies entirely above (or on) the horizontal line $y = a$.
- The number $a$ is called a lower bound of the function.
- The function never dips below this "floor."
Example: $g(x) = \sqrt{x} - 1$ on $[0, +\infty)$
At $x = 0$: $g(0) = \sqrt{0} - 1 = -1$
As $x$ increases, $g(x)$ increases without limit.
The minimum value is $-1$, so $g(x) \geq -1$ for all $x \in [0, +\infty)$.
This function is bounded below with lower bound $a = -1$.
At $x = 0$: $g(0) = \sqrt{0} - 1 = -1$
As $x$ increases, $g(x)$ increases without limit.
The minimum value is $-1$, so $g(x) \geq -1$ for all $x \in [0, +\infty)$.
This function is bounded below with lower bound $a = -1$.
Interactive visualization — available in the full lesson