At its essence, a function is a rule of correspondence that assigns each element from a set of inputs (the domain) to exactly one element in a set of outputs (the range). This deterministic relationship serves as the fundamental building block of mathematical modeling, allowing us to describe how one variable’s behavior is strictly dictated by another.
Consider a Salt Concentration Model: if we pump brine into a tank of pure water, the concentration $C(t)$ is a function of time $t$. For every specific moment we choose, there is only one possible concentration level. This "one-input, one-output" rule is the heart of calculus.
The Definition of a Function
A function $f$ is a rule that assigns to each element $x$ in a set $D$ exactly one element, called $f(x)$, in a set $E$. We represent this algebraically through formulas like:
- $y = mx + b$ (Linear)
- $f(x) = \sqrt{x}$ (Root)
- $\{(x, f(x)) \mid x \in D\}$ (Set-theoretic definition)
A function is not just a formula; it can be defined by a table of values (a tabular function) or even just a set of ordered pairs.
The Vertical Line Test (VLT): A curve in the $xy$-plane represents a function of $x$ if and only if no vertical line intersects the curve more than once. This ensures the "single-output" requirement is met.
Practical Evaluation: The Difference Quotient
To measure change in these relationships, we often evaluate the expression $\frac{f(a+h) - f(a)}{h}$.
Let $f(x) = 2x^2 - 5x + 1$. To evaluate the difference quotient:
- Substitute $(a+h)$ into $f$: $f(a+h) = 2(a+h)^2 - 5(a+h) + 1$
- Expand: $2(a^2 + 2ah + h^2) - 5a - 5h + 1 = 2a^2 + 4ah + 2h^2 - 5a - 5h + 1$
- Subtract $f(a)$: $(2a^2 + 4ah + 2h^2 - 5a - 5h + 1) - (2a^2 - 5a + 1) = 4ah + 2h^2 - 5h$
- Divide by $h$: $\frac{4ah + 2h^2 - 5h}{h} = 4a + 2h - 5$.