Functions notations, terminology, key concepts

Before functions, you were solving individual equations. Functions allow us to study groups or families of equations, in a way that makes algebra less abstract and more intuitive.

Contents

• Notation
• Definitions and basic properties
  • increasing and decreasing
• Evaluating functions
• Why $x$?
• Bonus: completeness of real numbers

Notation

The main notation looks like $f(x) = x^2 + 1$, where some input $x$ is squared then added by $1$.

For purposes of giving examples of functions without having to name them, the ${x \mapsto x^2+1}$ notation is also used.

Definitions and basic properties

The following definitions are simplifications. More rigorous discussions may have very different definitions, or use different terms for the following definitions.

A function returns a fixed output for every input. Meaning that ${a = b \implies f(a) = f(b)}$.

A graph is a visual representation of a function, typically over the $xy$-plane. The graph of a function (of $x$) passes the vertical line test, in which each vertical line passes through at most one point of the graph.

Mapping refers to the relationship between inputs and outputs, which can be a function or a non-function relation (ie a single input associated with multiple outputs).

Coordinates such as $(3, -1)$ is the point where $x = 3$ and $y = -1$. The $x$-coordinate is $3$, and the $y$-coordinate is $-1$.

Domain is the set of possible inputs.

Range is the set of possible outputs.

$x$-intercepts, or zeros, are $x$-coordinates for points on the $x$-axis. $y$-intercept is the $y$-coordinate for points on the $y$-axis.

A composite function $h = (f \circ g)(x) = f(g(x))$ means the output of $g$ is used as the input to $f$. For example if $f(3) = 5$ and $g(2) = 3$, then ${f(g(2)) = f(3) = 5}$.

A function is continuous when its graph can be drawn by a single connected line or curve. A function is continuous at $x = a$ when the graph near and including $x = a$ can be drawn by a single connected line or curve. More on this at the bottom of this page.

A discontinuity exists at some input where the function is not continuous.

increasing and decreasing

An increasing function satisfies ${b > a \implies f(b) \geq f(a)}$.

A strictly increasing function satisfies ${b > a \iff f(b) > f(a)}$.

A decreasing function satisfies ${b > a \implies f(b) \leq f(a)}$.

A strictly decreasing function satisfies ${b > a \iff f(b) < f(a)}$.

See also notations#iff and calculus: increasing and decreasing functions.

A function is one-to-one and invertible when ${f(a) = f(b) \implies a = b}$, namely no two inputs return the same output. Example: ${f(x) = \e^x}$. A continuous, invertible function is either strictly increasing or strictly decreasing.

The inverse of an invertible function $f$ is labelled as $f^{-1}$, such that ${f(f^{-1}(x)) = f^{-1}(f(x)) = x}$.

In general, $f(x) = k$ is solved by $x = f^{-1}(k)$ for invertible $f$. When it is not invertible, split $f$ into invertible pieces, namely ones that are strictly increasing or strictly decreasing.

$f$ and $f^{-1}$ have domains and ranges swapped.

The graphs of $f(x)$ and $f^{-1}(x)$ are mirror images of each other across $y = x$ diagonal line.

An asymptote is a line that represents some limiting behavior.

A horizontal asymptote indicates the function approaches some constant for very positive or negative inputs. A vertical asymptote indicates the function approaches positive or negative infinity near some finite input.

A function can have multiple, in fact an infinite number of vertical asymptotes, but at most two horizontal asymptotes. A continuous function has no vertical asymptotes.

Evaluating functions

Functions can be evaluated at any expression. Suppose we have ${f(x) = 2x \e^x}$, then

In summary, to evaluate a function at a particular value or expression, replace all $x$ by the value or expression, and simplify if possible.

Why $x$?

In the good old days where system of linear equations was the hardest thing ever, $x$ (and other variables) only represented some value. Now, variables can represent any expressions, including expressions containing other variables.

In other words $x$ is just a placeholder.

In fact, $f(x) = (x + 2)(x + 3)$ and $g(k) = k^2 + 5k + 6$ are the same function, even though $g$ looks different and uses $k$ instead of $x$. Two functions are identical as long as they describe the same mapping with the same domain and range.

This also means $x\mapsto \frac{x}{x}$ and $x\mapsto 1$ are different functions, because the first one cannot be evaluated at $x = 0$ while the second one can.

So we could have chosen any symbols, but as a convention (ie custom), input is usually $x$, and output is usually $y$.

Bonus: completeness of real numbers

In SL, all function inputs and outputs are real numbers ($x \in \mathbb{R}$), though not necessarily all real numbers. The set of real numbers includes all numbers on the number line, so real numbers are “one-dimensional” as each has one piece of information.

One very surprising property is that all real numbers can be simultaneously mapped to any non-zero finite interval, without repeats or overlaps. For example, the function ${\arctan x}$ maps all real numbers to the interval ${\left]-\frac{\pi}2, \frac{\pi}2\right[}$ and no two numbers are mapped to the same number. Similarly, any finite interval of real numbers can map to the entire set of real numbers without gaps.

Real numbers are said to be complete , meaning they cover the entire number line. This formalizes what it means for real numbers to be continuous.