Numbers & Functions · 3 Functions as Machines

3.5 Writing it down properly: sets and intervals

So far I’ve described domains in plain English. Mathematicians need to write these down, precisely and compactly — and so will you. A few symbols you’ll see constantly: Symbol Read aloud as Means ≥ “greater than or equal to” at least this big ≤ “less than or equal to” at most this big >, < “greater than”, “less than” strictly bigger / smaller ∈ “is an element of” belongs to this collection R “the reals” the whole real number line A collection of numbers is called a set. There are two standard ways to write one. Set-builder notation describes the members by a rule: { x | x ≥0 } reads “the set of all x such that x is greater than or equal to zero.” Interval notation just names the two ends: [0, ∞) means “everything from 0 up to infinity.” But look closely at the brackets, because they carry real information: • A square bracket [ ] means the endpoint is included.

• A parenthesis ( ) means the endpoint is excluded.

And ∞always gets a parenthesis, never a square bracket — because infinity isn’t a number you can reach, it’s a direction. When a machine jams at a single point in the middle, like h(x) = 1 x at 0, we glue the two surviving pieces with the union symbol ∪: (−∞, 0) ∪(0, ∞).

[0, ∞) (0, ∞) (2, 5) 2 5 [4, ∞) (−∞, 0) ∪(0, ∞) 0 Figure 3.2. Interval notation at a glance. A filled dot (square bracket) includes the endpoint; an open dot (parenthesis) excludes it; ∞is always open. The last row shows a domain with a hole punched out at 0. Now we can write our machines down like grown-ups: • Squaring f(x) = x2: domain (−∞, ∞), range [0, ∞).

• Square-root g(x) = √x: domain [0, ∞), range [0, ∞).

• Reciprocal h(x) = 1 x: domain (−∞, 0) ∪(0, ∞), range the same.

Functions as Machines

3.6 Drawing the machine: graphs

There’s a beautiful way to see an entire function at once. Take your input x and your output f(x), and treat them as a pair — an address. The input tells you how far across to go; the output tells you how far up. Mark a dot there. Do that for every input, and the dots trace out a picture of the whole machine. x f(x) (0, 0) (1, 1) (2, 4) (-2, 4) two inputs → one output never dips below the axis range [0, ∞) Figure 3.3. The squaring machine’s graph: a parabola. You can read the range straight off it — the curve never dips below the axis, the visual form of “squaring never gives a negative.” Note two inputs (−2 and 2) landing on the same output (4).

The dip at the bottom sitting on 0 is the machine’s smallest output. The two arms climbing upward show that big inputs — positive or negative — produce big positive outputs. The graph isn’t decoration. It’s the whole machine, laid out flat so your eyes can take it in at a glance.

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