# Chapter 1: Functions, Graphs and Lines > The vocabulary of calculus: what a function is, how to read graphs, domain and range, inverses, composition, odd/even symmetry, linear functions and the menagerie of standard graphs. **Canonical:** https://calculus.kitzo.work/chapter/functions **Level:** Starter **Estimated time:** 45 minutes **Exercises:** 8 (auto-marked; answers stay server-side) ## Topics - Interval notation & domain - Range from graphs - Vertical line test - Inverse functions & the horizontal line test - Composition of functions - Odd and even functions - Linear functions & slope - Catalog of standard graphs **Why this matters** Everything in calculus — limits, derivatives, integrals — is just functions in disguise. If you can read a graph, undo a function, and chain two functions together, the rest of the course has solid ground under its feet. This chapter is that ground. ## 1 · What a function actually is A **function** is a machine with one iron rule: **each input gives exactly one output**. Feed it x, it hands back f(x) — always the same answer for the same x, never two different ones. The set of inputs we are allowed to feed the machine is the **domain**. The set of outputs it can produce is the **range**. ### Interval notation, in one breath `[a, b]` includes the endpoints, `(a, b)` excludes them, `[a, ∞)` runs forever to the right. The symbol ∪ means "union — either of these sets", and ∞ always gets a round bracket because you can never actually arrive there. **Worked example** **Example 1 — Find the domain of f(x) = √(x − 2).** Step 1: the square root of a negative number is not a real number, so we need x − 2 ≥ 0. Step 2: solve the inequality: x ≥ 2. Therefore the domain is `[2, ∞)`. At exactly x = 2 the output is 0 — perfectly legal, so the bracket is square. ## 2 · Reading graphs like a detective A graph is a picture of *all* the input–output pairs at once. Three interrogation techniques: - **Vertical line test** — slide a vertical line across the graph. If it ever stabs two points, it is not a function. One hit per vertical line = legitimate function. - **Range from a graph** — squash the graph flat against the y-axis and look at the shadow. The vertical extent of that shadow is the range. - **Intercepts** — where the graph breaks the x-axis (outputs of 0) and the y-axis (the output at x = 0). ## 3 · Inverse functions — running the machine backwards If f sends 4 → 9, the inverse f⁻¹ sends 9 → 4. Two functions are inverses when each one undoes the other: `f(f⁻¹(x)) = x` and `f⁻¹(f(x)) = x`. **The recipe:** write y = f(x), swap x and y, solve for y. Done. **Watch out** **f⁻¹(x) is not 1/f(x)!** The ⁻¹ is about undoing, not reciprocating. The reciprocal of f(x) is written [f(x)]⁻¹ or 1/f(x). A function only has an inverse when it is **one-to-one** — no output is produced by two different inputs. Test it with the **horizontal line test**: a horizontal line must not hit the graph twice. (This is also exactly the vertical line test applied to the inverse's graph.) **Restricting the domain.** f(x) = x² fails the horizontal line test, so it has no inverse as-is. But on `[0, ∞)` it passes, and then f⁻¹(x) = √x. Domain restriction is not a hack — it is the definition of arcsin, arccos and friends later in the course. **Worked example** **Example 2 — Find the inverse of f(x) = 2x + 6.** Step 1: write y = 2x + 6. Step 2: swap: x = 2y + 6. Step 3: solve: 2y = x − 6, so `f⁻¹(x) = (x − 6)/2`. Sanity check: f(−3) = 0 and f⁻¹(0) = −3. ✓ ## 4 · Composition — machines plugged into machines `(f ∘ g)(x)` means **f(g(x))** — do g first, then feed its output into f. Order matters: generally f(g(x)) ≠ g(f(x)). **Worked example** **Example 3 — g(x) = x², h(x) = x + 3. Find g(h(2)).** Step 1: work from the inside out. h(2) = 2 + 3 = 5. Step 2: feed that into g. g(5) = 5² = 25. Compare: h(g(2)) = h(4) = 7. Different — that is why notation like f ∘ g exists. ## 5 · Odd and even — symmetry shortcuts Check `f(−x)`: - **Even** — f(−x) = f(x). Mirror symmetry across the y-axis. (cos x, x², |x|) - **Odd** — f(−x) = −f(x). Rotational symmetry about the origin. (sin x, x³, x) - **Neither** — everything else. (x² + x) Why we care: odd/even functions make integrals over symmetric intervals vanish or double — a trick you will use for years. ## 6 · Linear functions — the heroes of calculus A linear function `f(x) = mx + b` has the same slope everywhere: `m = rise / run`. Every derivative in this course generalizes what the slope of a line does for triangles — the derivative *is* "slope of the tangent line", made rigorous. **Two points determine a line:** given (x₁, y₁) and (x₂, y₂), the slope is m = (y₂ − y₁)/(x₂ − x₁). Through (1, 2) and (3, 8): m = 6/2 = 3. ## 7 · The standard cast of graphs You should recognize these on sight: `y = x`, `y = x²`, `y = x³`, `y = √x`, `y = |x|`, `y = 1/x`, `y = 2ˣ`, `y = ln x`, `y = sin x`. Calculus questions are ninety-percent "do you recognize the graph" and ten-percent "algebra the graph". **Chapter formula sheet** f(f⁻¹(x)) = x and f⁻¹(f(x)) = x m = (y₂ − y₁)/(x₂ − x₁) (f ∘ g)(x) = f(g(x)) Inverse recipe: y = f(x) → swap → solve for y Vertical line test ⇒ function · Horizontal line test ⇒ inverse exists Even: f(−x) = f(x) · Odd: f(−x) = −f(x) · Domain first, always ## Common mistakes - Forgetting ∞ always wears parentheses: (−∞, 3], not [−∞, 3]. - Writing f⁻¹(x) to mean 1/f(x). Don't. - Claiming g has an inverse without restricting its domain (x²!). - Composing in the wrong order: f(g(x)) means g first. - Reading the range off a graph as "where it starts" — squash it onto the y-axis instead. ## Practice exercises 1. The domain of f(x) = √(x − 2) is: - A. x ≥ 2 - B. all real numbers - C. x > 2 - D. x ≤ 2 2. If f(x) = 3x² − 5 , compute f(4) . 3. Find the inverse of f(x) = 2x + 6 . Write f⁻¹(x) = … 4. A graph passes the vertical line test when: - A. it never crosses the x-axis - B. every vertical line meets it at most once - C. every horizontal line meets it at most once - D. it is always increasing 5. Let g(x) = x² and h(x) = x + 3 . Compute g(h(2)) . 6. Classify f(x) = x³ − x : - A. even - B. odd - C. neither odd nor even - D. both odd and even 7. What is the slope of the line through (1, 2) and (3, 8)? 8. Write the set {x : x > 3 or x ≤ −2} in interval notation. --- Practice interactively: https://calculus.kitzo.work/chapter/functions Citation: CalculusKitzo (https://calculus.kitzo.work/). Independent study notes inspired by the standard calculus syllabus; not affiliated with Princeton University Press. See https://calculus.kitzo.work/disclaimer