Chapter 2 — Key Takeaways

A structured recap of Functions and Models: the big ideas, the formulas and tables to know cold, the standard procedures, the errors to avoid, and the threads that connect this chapter to the rest of calculus.

The Big Ideas

1. A function is a rule, not a formula and not a graph. The formula $f(x) = x^2$, the parabola you draw, the Python def f(x): return x**2, and a table of values are four representations of one underlying rule. A function returns exactly one output per input. (§2.1)

2. A function's domain is part of its identity. The squaring rule on $\mathbb{R}$ and the squaring rule on $[0,\infty)$ are different functions, even though they share a formula — and only the second is invertible. (§2.1, §2.5)

3. The standard library is small. Polynomial, rational, power, exponential, logarithmic, and trigonometric functions — plus their transformations, piecewise gluings, compositions, and inverses — are essentially everything you will differentiate and integrate for 38 chapters. Know each family's shape, domain, range, and special values. (§2.2)

4. Transformations turn one known shape into a family. Read them off the graph instead of memorizing algebra: inside the function, everything runs backwards. (§2.3)

5. Piecewise functions are how the real world's rules actually look — taxes, tariffs, overtime pay, $|x|$. The interesting calculus lives at the seams. (§2.4)

6. Composition builds complexity; inverses undo it. Composition drives the chain rule (Chapter 7); inverses require one-to-one functions. (§2.5)

7. A function can be a model — a deliberate, simplified stand-in for a real relationship. Calculus is the toolkit that analyzes models once they exist. No model, no calculus. (§2.7)

8. In calculus, all angles are in radians, always. Degrees inject a stray factor of $\pi/180$ into every derivative. (§2.2)

Formulas and Facts Worth Memorizing

Growth-rate hierarchy (as $x \to \infty$, for any $a > 0$): $$\ln x \;\prec\; x^a \;\prec\; e^x.$$ Logarithms grow slower than every positive power; exponentials grow faster than every power.

Logarithm laws: $\ln(xy) = \ln x + \ln y$, $\quad\ln(x/y) = \ln x - \ln y$, $\quad\ln(x^a) = a\ln x$, $\quad e^{\ln a} = a$, $\quad \ln(e^a) = a$.

Trig special values:

Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$.

Domain/range table:

Standard Procedures

Find a natural domain (§2.1). Handle each restriction separately, then intersect: (i) no division by zero, (ii) no even root of a negative, (iii) no log of a non-positive number.

Apply a transformation (§2.3). For $g(x) = a\,f\!\bigl(b(x - h)\bigr) + k$, resolve inside-out: horizontal shift $h$ and horizontal scale/reflect by $b$ act on the input (backwards); then vertical scale/reflect by $a$ and vertical shift $k$ act on the output.

Compose (§2.5). $(f\circ g)(x) = f(g(x))$ — apply the inner function $g$ first.

Invert (§2.5). Confirm $f$ is one-to-one (horizontal line test); write $y = f(x)$, swap $x\leftrightarrow y$, solve for $y$; set the inverse's domain equal to $f$'s range.

Plot in Python (§2.6). linspace $\to$ vectorize $\to$ plot. Use np.where for piecewise functions, plt.semilogy to compare growth rates, and sympy to verify inverses and compositions symbolically.

Common Errors to Avoid

• Confusing the function with its formula. "The function" means the rule (and its domain), however it is written. (§2.1)
• Getting the horizontal shift backwards. $f(x - 3)$ shifts right; $f(x + 3)$ shifts left. (§2.3)
• Reading $f^{-1}$ as a reciprocal. $f^{-1}$ is the inverse function; $\sin^{-1}x = \arcsin x$, whereas $(\sin x)^{-1} = \csc x$. (§2.5)
• Inverting a non-one-to-one function without restricting the domain first (e.g. $x^2$, or $\sin x$). (§2.5)
• Working in degrees. Calculus runs in radians. (§2.2)
• Believing np.where avoids invalid branches. np.where(cond, a, b) evaluates both a and b everywhere before selecting, so an undefined branch can still emit a warning. (§2.6)

Skills You Should Now Have

• State the domain and range of any standard-library function.
• Sketch any function reachable from a familiar one by a sequence of transformations.
• Evaluate a piecewise function and reason about its seams.
• Compute compositions and inverses by hand, and verify them with sympy.
• Plot any function with matplotlib using the linspace → vectorize → plot pattern.
• Translate a real scenario into a candidate model and critique it against the four criteria (faithfulness, simplicity, predictiveness, tractability).

Connections — Where This Goes

• Chapter 3 (Limits) makes "approaches a value," asymptotes, and long-run behavior precise — the doorway into calculus proper.
• Chapter 4 (Continuity) dissects the seams of piecewise functions: continuous vs. cornered.
• Chapter 6 (The Derivative) measures the rate of change of these functions; Chapter 5 first frames rate of change geometrically.
• Chapter 7 (Differentiation Rules) rewards the composition-spotting of §2.5 with the chain rule, and proves $\frac{d}{dx}e^x = e^x$ and the sigmoid derivative identity.
• Chapter 13 (The Definite Integral) accumulates these functions.
• Chapter 19 (Differential Equations) upgrades the exponential growth model into the logistic and SIR models (Case Study 1).
• Chapter 30 (Gradient & ML) puts the sigmoid (Case Study 2) inside multivariable gradient descent.

A Closing Reflection

Before moving on, sit with one harder question: why is calculus mostly about functions, and not about sets or numbers?

Because calculus is about change, and a function is the natural object for representing change: it tells you, for each value of an input, the corresponding output — and the very shape of a function (its slope, concavity, growth) encodes the change you care about. Functions are not mere notation; they are the carriers of the information calculus operates on. Master functions, and everything that follows — limits, derivatives, integrals — has somewhere to stand.

Next: Chapter 3 introduces the limit, conceptually the most important idea in Part I. Read it slowly.