VCE Mathematical Methods

## Relations and functions A **relation** is a set of ordered pairs $(x, y)$. A **function** is a relation in which each $x$-value is paired with exactly one $y$-value. On a graph this is the vertical line test: a function is cut at most once by any vertical line. The graph of $y = x^2$ passes the test, while the graph of $x = y^2$ does not, because $x = 4$ gives both $y = 2$ and $y = -2$. A function is **one-to-one** if each $y$-value also comes from exactly one $x$-value, which on a graph means every horizontal line cuts it at most once. The function $y = x^3$ is one-to-one and $y = x^2$ is not. This matters later, because only a one-to-one function has an inverse function. ## Function notation A function is specified by its rule, its domain and its codomain, and is written $$ f: X \to Y, \qquad f(x) = \text{rule}. $$ The **domain** is the set of inputs, the **codomain** is the set the outputs are taken from, and the **range** is the set of outputs that actually occur. The range is always a subset of the codomain. For $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x^2$, the codomain is $\mathbb{R}$ but the range is $[0, \infty)$. Notation for sets of real numbers: | Notation | Meaning | |---|---| | $\mathbb{R}$ | all real numbers | | $\mathbb{R}^+$ | positive real numbers, $x > 0$ | | $\mathbb{R} \setminus \{a\}$ | all real numbers except $a$ | | $[a, b]$ | $a \le x \le b$ | | $(a, b)$ | $a < x < b$ | | $[a, b)$ | $a \le x < b$ | | $(a, \infty)$ | $x > a$ | A square bracket…