Boundaries of Reality
Every function has limits on what it can accept and what it can produce.
- Domain: The set of all possible inputs (\(x\)).
- Range: The set of all possible outputs (\(y\)).
The Two Domain Killers
In real-number algebra, there are two things you can never do:
- Divide by Zero: The denominator cannot be 0.
- Square Root of a Negative: The expression inside a radical must be \(\geq 0\).
Worked Examples
Example 1: Rational Function
Find the domain of \(f(x) = \frac{10}{x - 3}\).
- Set denominator to zero: \(x - 3 = 0 \to x = 3\).
- The function breaks at 3.
- Domain: All real numbers except \(x = 3\). Notation: \((-\infty, 3) \cup (3, \infty)\).
Example 2: Radical Function
Find the domain of \(f(x) = \sqrt{x + 5}\).
- Set inside \(\geq 0\): \(x + 5 \geq 0 \to x \geq -5\).
- Domain: \([-5, \infty)\).
Example 3: Finding Range
Find the range of \(f(x) = x^2 + 4\).
- \(x^2\) is always at least 0.
- Adding 4 means the smallest output is \(0 + 4 = 4\).
- Range: \([4, \infty)\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, the "Domain" is the physical space where a particle can exist. If a particle is inside a box of length \(L\), its wavefunction \(\psi(x)\) has a domain of \([0, L]\). Outside this domain, the probability is zero. The "Range" of the wavefunction's values is also restricted; for a particle to "exist," the total probability across the domain must add up to exactly 1. This is called Normalization. Understanding the limits of your functions is how you define the physical boundaries of an experiment.