Lecture · 14 min · basic

Functions

A rule, a graph, and a machine

Hardy, Pure Mathematics · Ch. II

A function is not “a formula with $x$ in it.” It is a pairing: every allowed input is sent to exactly one output. $f (x) = x^{2}$ and $g (t) = t^{2}$ are the same function. The letter is a dummy. The rule is the object.

On the board today

—Distinguish a function from an expression.
—Read domain and range from a rule and from a graph.
—Evaluate compositions $f \circ g$ .

Composition

(f \circ g) (x) = f (g (x))

Apply $g$ first, then $f$ . Order matters.

The graph of $f$ is the set of points $(x, f (x))$ . The vertical-line test is the uniqueness clause in disguise: a vertical line is “one input,” and it may hit the graph at most once. Domain is the set of legal inputs; range is the set of outputs that actually appear.

Worked example · Compose two rules

Let $f (x) = 2 x + 1$ and $g (x) = x^{2}$ . Compute $(f \circ g) (3)$ and $(g \circ f) (3)$ .

1. $(f \circ g) (3) = f (9) = 19$ .
2. $(g \circ f) (3) = g (7) = 49$ .

Composition is not commutative. $19 \neq = 49$ .

Pause the lecture

Is $y^{2} = x$ the graph of a function $y = f (x)$ ?

Common mistake. $f (x + h)$ is not $f (x) + h$ and not $f (x) + f (h)$ in general. Substitute the entire input into the rule. This mistake wrecks difference quotients.

Practice this lecture

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