Functions and Their Graphs

What a function really is#

A function is a machine with a guarantee: feed it an input from its domain, and it returns exactly one output. Write f(x) = x^2 and you have described infinitely many facts at once — f(3)=9, f(-3)=9, f(\pi)=\pi^2 — all compressed into one rule.

The graph of f is the set of points (x, f(x)): a complete portrait of the function’s behavior. Reading graphs fluently is the single most valuable skill in this book, because a graph converts algebra (hard to feel) into geometry (easy to see).

• Domain: the inputs allowed. For f(x)=\sqrt{x}, the domain is x \ge 0; for f(x) = \frac{1}{x-2}, everything except 2.
• Range: the outputs actually achieved.
• Vertical line test: a curve is the graph of a function exactly when every vertical line meets it at most once — one input, one output.

Straight lines#

The straight line y = mx + c is the simplest function with something to say. The slope m is the exchange rate between y and x: every unit step right raises the graph by m. The intercept c is where the line crosses the y-axis.

Two forms cover every situation:

\begin{aligned}\text{slope–intercept: } y = mx + c \\ \text{point–slope: } y - y_1 = m(x - x_1).\end{aligned}

Given two points, the slope is m = \dfrac{y_2 - y_1}{x_2 - x_1} — rise over run. Parallel lines share slopes; perpendicular lines have slopes multiplying to -1 (each is the other’s negative reciprocal).

Transformations: every graph is an old friend, moved#

Once you know one graph, you know a whole family. For any function f:

The inside-the-brackets moves feel backwards (f(x-2) moves right) — the resolution is that changing the input relabels the x-axis: to get the old value f(0), you must now feed in x = 2.

Combined transformations are read in a definite order, and getting it wrong is the commonest source of a wrong sketch. Take y = -2(x+1)^2 + 3, built from y = x^2:

\begin{aligned}y = x^2 &\longrightarrow (x+1)^2 \\ &\quad \text{inside first: } +1 \text{ shifts left } 1 \text{ — backwards, as always} \\ &\longrightarrow 2(x+1)^2 \\ &\quad \text{stretch vertically by } 2 \\ &\longrightarrow -2(x+1)^2\end{aligned}

the minus flips it upside down

\begin{aligned}&\longrightarrow -2(x+1)^2 + 3 \\ &\quad \text{finally lift } 3\end{aligned}

The result is a downward parabola with vertex (-1, 3). The rule to hold on to: inside the bracket, act first and act backwards; outside the bracket, act last and act as written. Every combined transformation in this book obeys it.

Composition and inverses#

Composition chains machines: (f \circ g)(x) = f(g(x)) — g acts first. Order matters: with f(x)=x^2 and g(x)=x+1, f(g(2)) = 9 but g(f(2)) = 5.

The inverse f^{-1} runs the machine backwards: f^{-1}(y) is the input that produced y. It exists precisely when f never sends two inputs to one output (one-to-one). Graphically, f^{-1} is f reflected across the line y = x — inputs and outputs swap seats. We will meet the most important inverse pair of all, e^x and \ln x, in Chapter 5.

In the wild. “Function” is the load-bearing idea of the modern world. A program’s function f(input) -> output is the mathematical concept, verbatim. An entire trained neural network is one function \hat y = f(\vec x;\theta) from inputs to predictions (Part IV makes this precise), and composition — Chapter 3’s f \circ g — is exactly how its layers stack. Domains matter in practice too: feed a function an input outside its domain and you get the software version of \frac{1}{0} — a crash.

If you keep one thing from this chapter: A function is a machine and its graph is the machine’s portrait; every graph you will ever meet is a familiar one, shifted, stretched, or flipped.