Relations and Functions

A relation is the object underneath

Before a function there is a relation. A relation between two variables $x$ and $y$ is simply a set of ordered pairs $(x, y)$. The set $R = \{(1, 5), (2, 5), (3, 7)\}$ is already a relation, all by itself: three points, nothing more. Reading $x$ as the input, $R$ pairs $1$ with $5$, $2$ with $5$, and $3$ with $7$. A repeated output like the two $5$s is allowed; the only thing that would cause trouble is a repeated input, and no input repeats here, so $R$ is a function of $x$. Read $y$ as the input instead and the pairs read $5 \to 1$, $5 \to 2$, and $7 \to 3$: now the input $5$ has two different outputs, $1$ and $2$, so $R$ is not a function of $y$. Same three points, opposite verdict.

An equation in $x$ and $y$ is a second, usually infinite, way to name a relation. The equation carves out of the whole plane exactly those points whose coordinates make the equation true, and that set of points is the relation. The equation $y = 2x + 1$ names a line, the equation $x^2 + y^2 = 1$ names a circle, and the equation $x = y^2$ names a sideways parabola. In every case the set of points comes first; the equation is just one way to describe it.

A function is not a new kind of object. It is a relation with one extra property laid on top, and we lay it there for a definite reason. We want to be able to evaluate, to feed in a value and get a single answer back. A relation lets a point sit anywhere; a function insists that once you fix the input, the output is pinned down with no ambiguity. So a function is a relation in which each input is paired with exactly one output. Everything in this lesson comes from taking that sentence apart, and the first surprise is hiding in the word input.

The same curve, two different questions

Which variable is the input? Nothing in a set of points answers that; it is a choice you make. And the whole verdict “function or not” can flip depending on the choice. The cleanest example is the sideways parabola $x = y^2$.

Read it with $x$ as the input. Ask for the output at $x = 4$: you need $y$ with $y^2 = 4$, and there are two, namely $y = 2$ and $y = -2$. One input, two outputs, so with $x$ as the input this relation is not a function. Now read the identical set of points with $y$ as the input instead. Ask for the output at any value of $y$: the rule $x = y^2$ hands back the single number $y^2$, one output every time. With $y$ as the input the very same relation is a function. The points on the page never moved. Only the question changed.

This is why the vertical line test is worth restating carefully. It is often remembered as a fact about the shape of a curve, but it is not. It is the test for whether reading $x$ as the input produces a function, and it is exactly the picture of “one output per input.”

Since nothing in the proof was special to $x$, swap the roles of the two variables and the same argument runs word for word. Reading $y$ as the input is a function if and only if no horizontal line meets the relation more than once. The horizontal line test is not a second rule to memorize; it is the vertical line test asked about the other variable, applied to the very same points. Nothing about the graph has to move to ask this new question.

A different picture appears only if you go one step further and redraw the relation with $y$ placed on the horizontal axis, the normal spot for whichever variable is the input. That redrawing swaps the two coordinates of every point: $(x, y)$ becomes $(y, x)$. On the page, swapping every point’s coordinates like that reflects the whole graph across the line $y = x$, and that reflection turns every vertical line into a horizontal one. So the new, redrawn graph passes the vertical line test exactly when the original graph passed the horizontal line test. This reflected, coordinate-swapped picture is the inverse relation. A later lesson will call it the inverse function exactly when the original relation was one-to-one, with its codomain taken to be its range: that is precisely the case just established, where the redrawn graph passes the vertical line test. So the question that opens this chapter, “can this rule be undone,” is underneath just the horizontal line test on the original graph, redrawn so the new input sits where an input belongs. We return to it in the inverse-functions lesson.

Two more relations are worth holding next to $x = y^2$. The unit circle $x^2 + y^2 = 1$ fails both tests. At $x = 0$ it has the two points $(0, 1)$ and $(0, -1)$, and at $y = 0$ it has $(1, 0)$ and $(-1, 0)$, so the circle is a function of neither variable. The cubic $y = x^3$ passes both. Each $x$ gives the single output $x^3$, and each $y$ comes from the single input equal to its cube root, so the cubic is a function of both. “Function” really does live in the pairing of a curve with a choice of input, not in the curve alone.

The domain is part of the function, not a leftover

Here is the second idea, and it is easy to miss because in earlier courses it never came up. A function is not just a rule. Fully specified, it is three pieces of data at once: a domain, the set of allowed inputs, and a rule, telling you what to do to an input. The third piece is a codomain, a set you declare the outputs to fall inside. Change any one of the three and you have changed the function. The rule is the piece students notice, because it is the formula you can write down. The domain is just as much a part of the function, and forgetting it is the source of the trouble in this section.

In earlier courses “the domain” always meant one specific thing without anyone saying so: the largest set of inputs on which the formula returns a real number. That set has a name, the natural domain of the formula (also called the implied domain). It is a convention, the domain you assume when no other is stated. The natural domain is found the way you already know: throw out inputs that divide by zero or take an even root of a negative. For $f(x) = \dfrac{1}{x - 3}$ the natural domain is every real number except $3$, and for $g(x) = \sqrt{x}$ it is every real number greater than or equal to $0$.

A declared domain is different. It is a domain stated outright as part of the function, and it can be smaller than the natural domain. Writing “let $f(x) = x^2$ for $x \ge 0$” declares the domain to be the nonnegative numbers, even though the formula $x^2$ would happily accept every real number. The formula does not force this restriction; the author of the function chooses it, and it travels with the function from then on. A declared domain can only shrink the natural domain, never grow past it: you could not declare "$x \ge 0$" as the domain of $f(x) = \dfrac{1}{x - 3}$, because that set still contains $x = 3$, an input where the formula has no value. A legal declared domain has to leave out $x = 3$ too, for instance ”$x \ge 0$ and $x \ne 3$”.

Intro algebra only ever computed the natural domain and called it “the” domain, which quietly assumed the domain is always read off the formula. It is not. The natural domain is a default, and a declared domain overrides it.

A declared domain does not have to come from algebra at all; it can come from what the numbers mean. Let $C(n) = 8n$ be the cost of $n$ tickets at 8 dollars each. The formula $8n$ is happy to accept any real number, but a ticket count must be a whole number and cannot be negative, so the situation declares the domain to be $n = 0, 1, 2, 3, \dots$, a much smaller set than the formula alone would allow.

You have already felt the domain acting as real data. In the Solving Linear Equations lesson of Chapter 1, clearing denominators in $\dfrac{x}{x-2} = \dfrac{2}{x-2}$ produced the candidate $x = 2$. That candidate had to be thrown out, because $x = 2$ was never in the domain of the original equation. The exclusion did not come from the algebra after clearing; it came from the domain, which was fixed before any step was taken. That is the same lesson in a different costume: the domain is decided up front and constrains everything that follows.

Same formula, different domain, different function

Because the domain is genuine data, two functions can share a formula letter for letter and still be different functions. This is not a technicality; it is the hinge the last lesson of this chapter turns on.

Compare $f(x) = x^2$ with domain all real numbers against $g(x) = x^2$ with domain $x \ge 0$. Same rule, different domains. They behave differently in a way that matters. The function $f$ is not one-to-one: $f(-2) = 4$ and $f(2) = 4$, so two different inputs share an output, and the horizontal line $y = 4$ meets its graph twice. The function $g$ is one-to-one: on the inputs $x \ge 0$, if $g(a) = g(b)$ then $a^2 = b^2$, and since $a$ and $b$ are both nonnegative this forces $a = b$. No horizontal line therefore meets the graph of $g$ more than once. Take each function’s codomain to be its range, so that “undone by a rule” is a fair question and not a mismatch of sets. Under that reading, $g$ can be undone by a rule and $f$ cannot, purely because their domains differ: one-to-one is exactly the property that lets every output be traced back to a single input.

Range and codomain

The output side has its own pair of words, and keeping them apart avoids a common muddle. The codomain is the set you declare the outputs to live in when you set the function up. The codomain is often just “the real numbers,” for lack of a reason to say anything smaller. The range is sharper: it is the set of outputs the function actually produces, every value $f(x)$ hit by some input in the domain.

The range always sits inside the codomain. The two sets need not be equal, because declaring a set for the outputs to live in does not promise that every element of it gets used. Take $f(x) = x^2$ on all real numbers with codomain the real numbers. Every output is a square, and a square is never negative, so the range sits inside the nonnegative numbers. It is not smaller than that either: for any target $y \ge 0$, the real input $x = \sqrt{y}$ gives $f(x) = y$, so every nonnegative number really is produced. The range is exactly that nonnegative set, a strict part of the real-number codomain: the negative numbers sit in the codomain but are never outputs. When a problem asks for “the range,” it is asking what the function really hits. Answering that takes a moment’s thought about the rule, not just a glance at where you declared the outputs.

A bound can belong to the range or sit just outside it, and the two cases look alike until you check. Take $f(x) = \dfrac{1}{x^2 + 1}$ on all real numbers. The denominator is at least $1$, so every output lies in $0 < f(x) \le 1$. The upper bound $1$ is attained: it is the genuine output at $x = 0$, since $f(0) = 1$. Every value strictly between $0$ and $1$ is attained too: given a target $0 < y \le 1$, solving $\dfrac{1}{x^2+1} = y$ for $x$ gives $x = \sqrt{\dfrac{1}{y} - 1}$, a real number because $y \le 1$ keeps $\dfrac{1}{y} - 1$ nonnegative. The lower bound $0$ is only approached: as $x$ grows the outputs shrink toward $0$, but a ratio of positive numbers is never $0$, so no input ever produces it. A value belongs to the range exactly when some input actually produces it. That is why the range is $0 < y \le 1$, closed at the attained end and open at the approached end. Deciding this is never automatic: for each candidate bound you check whether it is a real output or only a value the outputs crowd against.

The rule and domain as a set of ordered pairs

Both ideas of this lesson fold into one description. A function’s rule and domain together are a set of ordered pairs in which no two different pairs share the same first coordinate. Declaring a codomain on top of those pairs completes the function. Read that description once for each idea.

Read for the first idea: which variable you treat as the input decides which pairs you write down. If $x$ is the input, use the pairs $(x, y)$ exactly as the relation gives them, and the rule against a repeated first coordinate is precisely the vertical line test. If $y$ is the input instead, swap the two entries of every pair to get $(y, x)$; whether that new set of pairs is again a function is precisely the horizontal line test, and the swapped set is the seed of the inverse.

Read for the second idea, the domain is not computed from anything. It is simply the set of first coordinates that appear among the pairs, so it is built in from the start; include different pairs and you have a different function. The range, in the same breath, is the set of second coordinates that appear. So the pairs carry the rule, the domain, and the range at once. The one piece they do not carry is the codomain, the larger set you declare the outputs to live in. So you state the codomain separately, and the function is then complete. Two ideas, one object.

Common mistakes