Functions

A function is a rule that assigns exactly one output to each input. We write $f(x)$ (read "f of $x$") for the output when the input is $x$.

On a graph, this means no vertical line crosses the graph more than once (the vertical line test). A circle fails the test: at $x = 0$ it has two $y$-values.

Function Rules

A function can be given by a formula, a table, a graph or a description in words. With a formula, evaluate by substituting the input everywhere $x$ appears:

$$f(x) = x^2 - 3x + 2 \qquad f(4) = 16 - 12 + 2 = 6$$

The input can be an expression too. For $g(x) = 2x + 1$:

$$g(a + 3) = 2(a + 3) + 1 = 2a + 7$$

Domain and Implied Domain

The domain is the set of allowed inputs. When a formula is given with no domain stated, its implied domain is every real number that doesn't break the formula. Watch for three things:

1. Division by zero. $\dfrac{1}{x - 5}$ needs $x \neq 5$, so its domain is $(-\infty, 5) \cup (5, \infty)$.
2. Even roots of negatives. $\sqrt{x - 2}$ needs $x - 2 \ge 0$, so its domain is $[2, \infty)$.
3. Logarithms of zero or negatives. $\ln x$ needs $x > 0$ (section 02.10).

Range

The range is the set of outputs the function actually produces. Since $x^2 \ge 0$ for every $x$, the function $f(x) = x^2 + 1$ has outputs $\ge 1$, and every such value is reached. Its range is $[1, \infty)$.

A graph makes ranges easy: look at how far up and down it reaches.

Injective and Surjective

• Injective (one-to-one): different inputs always give different outputs. On a graph, no horizontal line hits it twice. $2x + 3$ is injective; $x^2$ is not, since $f(-2) = f(2)$.
• Surjective (onto): every value in the target set (the codomain) is an output. $x^2$ from $\mathbb{R}$ to $\mathbb{R}$ is not onto, because it never outputs $-1$. From $\mathbb{R}$ to $[0, \infty)$ it is.
• Bijective: both. Bijective functions are exactly the ones with inverses (section 02.04).