Functions as Input-Output Relationships, foundations for calculus lesson

A vending machine is a fair picture of a function. You press a button and one particular snack drops out, the same snack every time you press that button. Two buttons may give the same snack, and that is fine. A button that sometimes gives chips and sometimes gives a candy bar would be broken. Calculus studies how outputs change as inputs change, and that only makes sense when each input has one definite output.

Key terms

Function

Definition A function $f$ from a set $D$ to a set $Y$ is an assignment that gives to every element $x$ of $D$ exactly one element of $Y$ . In this course $D$ and $Y$ are sets of real numbers unless stated otherwise. Two conditions are built into this definition. Every element of $D$ gets an output, and no element of $D$ gets more than one.

In plain words A function takes each allowed input and hands back one output, never zero and never two.

Example On the inputs $\{1, 2, 3\}$ , the assignment $1 \mapsto 5$ , $2 \mapsto 7$ , $3 \mapsto 5$ is a function. The assignment $1 \mapsto 5$ , $1 \mapsto 6$ , $2 \mapsto 7$ is not, for two reasons. The input $1$ gets two different outputs, and the input $3$ gets none.

Input and output

Definition An input of a function is an element $x$ of its domain. The output at $x$ is the single element that the function assigns to $x$ , written $f(x)$ and also called the value of $f$ at $x$ . The set of all outputs, $\{f(x) \mid x \in D\}$ , is the range of $f$ .

In plain words The input is what you put in, the output is what comes out. In a real situation each comes with its own meaning and units.

Example Let $T(m)$ be the temperature in degrees Celsius $m$ minutes after a cup of tea is poured. The input $m = 10$ is a time in minutes. If $T(10) = 62$ , the output $62$ is a temperature in degrees Celsius. Mixing the two up, for example reading $T(10) = 62$ as “after 62 minutes”, misreads the function.

Domain

Definition The domain of a function is the set of inputs on which it is defined. It is part of the function and should be stated. When a function is given only by a formula and no domain is stated, the domain is taken to be the set of all real numbers for which the formula produces a real number. A context, such as time or length, may restrict the domain further.

In plain words The domain is the list of inputs the function accepts. Change the domain and you have changed the function.

Example The formula

r(x) = \frac{1}{x - 2}

has domain $(-\infty, 2) \cup (2, \infty)$ , since $x = 2$ makes the denominator zero. So $r(4) = \frac{1}{2}$ and $r(0) = -\frac{1}{2}$ , but $r(2)$ does not exist. If the same formula described a quantity at time $x$ hours after noon with $0 \le x \le 1$ , the domain would be $[0, 1]$ .

Rule

Definition A rule is a description, usually a formula, that tells how to compute the output from the input. A rule defines a function only together with a domain on which it produces exactly one output for each input. The same rule on different domains gives different functions, and a function may also be given by a rule in words or by several formulas on different parts of its domain.

In plain words The rule is the recipe. The domain says which ingredients the recipe is meant for.

Example The rule “square the input and subtract $1$ ” is the formula $g(x) = x^2 - 1$ . On the domain $(-\infty, \infty)$ it gives

\begin{gathered} g(-2) = (-2)^2 - 1 = 3, \\[0.4em] g(0) = -1, \qquad g(3) = 8. \end{gathered}

Restricted to the domain $[0, \infty)$ , the same rule makes a different function, which no longer accepts $-2$ .

Function notation

Definition The symbol $f(x)$ denotes the output of the function $f$ at the input $x$ . The letter $f$ names the function, and the parentheses enclose the input. The notation does not mean $f$ multiplied by $x$ . The input may be a number or any expression in the domain, as in $f(3)$ , $f(a + h)$ or $f(2x)$ , and evaluating means substituting that whole input into the rule.

In plain words Read $f(x)$ as “f of x”, the output when $x$ goes in. You cannot cancel or distribute the $f$ .

Example For $f(x) = x^2 + 1$ ,

\begin{gathered} f(3) = 10, \\[0.4em] f(2x) = (2x)^2 + 1 = 4x^2 + 1, \\[0.4em] 2f(x) = 2x^2 + 2. \end{gathered}

So $f(2x)$ and $2f(x)$ are different. Also $f(1 + 2) = f(3) = 10$ , while $f(1) + f(2) = 2 + 5 = 7$ , so you cannot split $f(1 + 2)$ as if $f$ were a number multiplying a sum.

Table

Definition A table of values lists inputs in one column or row and the corresponding outputs beside them. A table describes a function exactly when each listed input appears with only one output. A finite table gives the values at the listed inputs only. Values at other inputs are not determined by the table unless a rule or an assumption supplies them.

In plain words A table is a list of input and output pairs. It is a function if no input appears with two different outputs.

Example The rule $g(x) = x^2 - 1$ produces this table.

$x$	$-2$	$-1$	$0$	$1$	$2$
$g(x)$	$3$	$0$	$-1$	$0$	$3$

The rule, the table and a picture of arrows from each input to its output all describe the same assignment on these five inputs.

Unique output

Definition Each input of a function has a unique output. For a relation given as a set of input and output pairs, this means that whenever $(x, y_1)$ and $(x, y_2)$ are both pairs with the same input $x$ , the outputs agree, $y_1 = y_2$ . A relation that pairs some input with two different outputs is not a function. The reverse is not required, so different inputs may share an output, and $f(x_1) = f(x_2)$ with $x_1 \ne x_2$ is allowed.

In plain words One input, one output. Two inputs landing on the same output is fine. One input pointing to two outputs is not.

Example In the table of $g(x) = x^2 - 1$ the inputs $-1$ and $1$ share the output $0$ , and $g$ is still a function. Now look at this table.

Input	$0$	$1$	$1$	$2$
Output	$4$	$5$	$6$	$7$

The input $1$ appears with the outputs $5$ and $6$ , so this table does not define a function. If the second $1$ had output $5$ again, it would only repeat the same pair and cause no conflict.

Check yourself

For $k(x) = \frac{x + 1}{x - 3}$ , find the domain, $k(5)$ and $k(-1)$ .

Show the answer

The denominator is zero at $x = 3$ , so the domain is $(-\infty, 3) \cup (3, \infty)$ . The values are

\begin{gathered} k(5) = \frac{6}{2} = 3, \\[0.4em] k(-1) = \frac{0}{-4} = 0. \end{gathered}

Does the assignment $2 \mapsto 0$ , $4 \mapsto 0$ , $6 \mapsto 1$ define a function on $\{2, 4, 6\}$ ? What about $2 \mapsto 0$ , $2 \mapsto 1$ , $6 \mapsto 1$ on $\{2, 6\}$ ?

Show the answer

The first is a function, because each input has one output, and sharing the output $0$ is allowed. The second is not, because the input $2$ is sent to both $0$ and $1$ .

For $f(x) = 3x - 2$ , compute $f(4)$ and $f(x + 1)$ , and explain why $f(4)$ does not mean $f$ times $4$ .

Show the answer

The values are $f(4) = 3 \cdot 4 - 2 = 10$ and $f(x + 1) = 3(x + 1) - 2 = 3x + 1$ . The notation $f(4)$ means the output when the input is $4$ , a single number, not a product of $f$ and $4$ .

Functions as Input-Output Relationships

Key terms

Function

Input and output

Domain

Rule

Function notation

Table

Unique output

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus