Input vs Output
Instead of writing \(y = 3x + 1\), we often write \(f(x) = 3x + 1\). This notation emphasizes that the function is a Machine: you put an input (\(x\)) into the machine, and it spits out an output (\(f(x)\)).
Pronunciation: \(f(x)\) is read as "f of x". It does not mean \(f \times x\).
Worked Examples
Example 1: Evaluating Functions
If \(f(x) = x^2 - 4\), find \(f(5)\).
- Replace every \(x\) with 5.
- \(f(5) = 5^2 - 4 = 25 - 4 = 21\).
- Result: 21
Example 2: Variable Input
If \(g(x) = 2x + 10\), find \(g(a + 3)\).
- Replace \(x\) with the entire expression \((a + 3)\).
- \(g(a + 3) = 2(a + 3) + 10 = 2a + 6 + 10 = 2a + 16\).
- Result: \(2a + 16\)
Example 3: Working Backward
If \(f(x) = 4x - 2\), find \(x\) such that \(f(x) = 14\).
- Set the output to 14: \(14 = 4x - 2\).
- Solve for \(x\): \(16 = 4x \to x = 4\).
- Result: 4
The Bridge to Quantum Mechanics
In Quantum Mechanics, the most important "function" is the Wavefunction \(\psi(x)\). This function tells us the "state" of a particle at any point in space. We also use Operators, which are like functions that act on other functions. For example, the energy operator \(\hat{H}\) "acts" on \(\psi(x)\) to give us the energy levels. Function notation is the foundational grammar for describing how physical systems change over time and space.