Notation and terminology

Unit 0.1 Notation and terminology

There are several ways to conceive of a function. One is that it is a rule that takes an input and gives you an output. This is how most of us think of functions most of the time, but it is not precise (rules are sentences which may be ambiguous or underspecified). For this reason we also need a formal definition. A third way to understand functions is via their graphs. We now discuss all three of these ways of characterizing a function, beginning with the most formal.

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Definition 0.1.

A function is a set of ordered pairs with the property that no two ordered pairs have the same first element. The domain of \(f\) is defined to the set of all first elements of the ordered pairs. The range of \(f\) is defined to be the set of all second elements of the ordered pairs.

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The expression \(f(x)\) is defined to equal to \(y\) if the ordered pair \((x,y)\) is in the set of ordered pairs defining \(f\) and undefined otherwise. Informally, \(f(x)\) is called the value of the function \(f\) evaluated at the argument \(x\text{.}\)

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Now let’s say the same things verbally. The domain of a function is the set of allowed inputs; the range is the set of all outputs. We often name functions with letters; \(f\) is the typical choice, then \(g\) if another is needed, but of course we could name a function anything. While it is common to refer to the function \(f\) as \(f(x)\text{,}\) we will try to observe the distinction that \(f\) is the function and \(f(x)\) is its value at the argument \(x\text{,}\) meaning the output when you plug in \(x\) as an input. The condition that first coordinates are distinct corresponds to the rule producing an unambiguous answer.

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Finally, to describe the function \(f\) via its graph, we interpret the ordered pairs as points in the plane, and draw this set as a curve. The condition that first coordinates are distinct corresponds to the so-called vertical line test: any vertical line (vertical lines being sets with a single fixed \(x\)-coordinate but all possible \(y\)-coordinates) intersects the graph at most once.

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Why be so formal?

In common usage, one might encounter any of the three ways of defining or referring to a function. We don’t want to drown in formality, so we usually use something only as formal as needed. Let’s look at why we sometimes need formality.

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Example 0.2.

Suppose we define a function \(f\) by \(f(x) := x^2 + 2\text{.}\) Have we formally defined this function? It sounds as if this is the set of ordered pairs

\begin{equation*} \{ \ldots , (-2, 6) , (-1, 3) , (0,2) , (1, 3) , (2 , 6) , \ldots \} . \end{equation*}

That would be if we meant the domain to be the set of all integers. Maybe instead we meant the domain to be the set of all real numbers. In that case, the "\(\ldots\)" in the list is somewhat misleading; we should probably write the set of ordered pairs like this:

\begin{equation*} \{ (x , x^2 + 2) : x \in \mathbb{R} \} \end{equation*}

(we use the notation \(\mathbb{R}\) for the real numbers and \(\in\) for the "is an element of"). If this function arose in a word problem where \(f(x)\) represented the value of some quantity at a time \(x\) seconds after the start, maybe it makes sense to allow only nonnegative real numbers as inputs. Formally, this would look like

\begin{equation*} \{ (x , x^2 + 2) : x \text{ is real and nonnegative} \}, \end{equation*}

which could also be written

\begin{equation*} \{ (x , x^2 + 2) : x \in [0,\infty) \} \end{equation*}

\begin{equation*} \{ (x , x^2 + 2) : x \geq 0 \}, \end{equation*}

this last version assuming we understood this to mean real numbers at least zero rather than, say, integers at least zero.

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A consequence of this discussion is that two functions are technically different if they have different domains, even if they have the same defining rule. We mostly don’t care about this distinction unless it matters to some problem, such as trying to determine the number of solutions to an equation, as in the next exercise.

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Technically, our discussion of the function \(x \mapsto x^2 + 2\) referred to three different functions: one whose domain was all integers, one whose domain was all reals, and one whose domain is all nonnegative reals. You can see they are different functions: even thought the defining equation \(f(x) := x^2 + 2\) is the same for all three, they are defined by different sets of ordered pairs. On the other hand, for many purposes, we don’t care which of these functions was intended. We can feel free to define the function by \(f(x) := x^2 + 2\) without specifying the domain unless and until we get into trouble with the ambiguity in the domain. If we try to answer a question like "How many solutions are there to \(f(x) = 3\text{?}\)" then we will need to be more precise about the domain.

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Checkpoint 6.

What are the respective numbers of solutions to \(f(x) = 3\) when \(f (x) := x^2 + 2\) and the domain is. . .

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the integers \(\left\{\ldots,-3,-2,-1,0,1,2,3,\ldots\right\}\)
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the real numbers \(\mathbb{R}\)
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the nonnegative reals \(\left\{x\in\mathbb{R}: x\geq 0 \right\}\)
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Hint.

This question doesn’t ask about what the solutions are; it only asks how many solutions there are.

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Answer 1.

\(2\)

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Answer 2.

\(2\)

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Answer 3.

\(1\)

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In the discussion so far, we have introduced four notations you are probably familiar with, but to be completely explicit, we discuss each briefly.

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Maps-to notation. 🔗: Often we name a function when defining it, then refer to it by name, but we can also refer to it using the "maps-to" symbol \(\mapsto\text{.}\) Thus, \(x \mapsto x^2 + 2\) refers to the function that we named \(f\text{,}\) above. We use this when mentioning a function but rarely when evaluating it at an argument because the notation \((x \mapsto x^2 + 2) (3)\) is an atrocity (but technically equal to 11).
Open and closed interval notation. 🔗: The interval \([a,b]\) refers to all real numbers \(x\) such that \(a \leq x \leq b\text{.}\) When both endpoints are included, this is called a closed interval.
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The interval \((a,b)\) refers to all real numbers \(x\) such that \(a \lt x \lt b\text{.}\) When both endpoints are excluded, this is called an open interval.
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Warning 0.3.

The notation for an open interval is exactly the same as for an ordered pair! If there is any ambiguity we will try to specify which, for example, "Let \((a,b)\) be the open interval..."
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The notations \((a,b]\) and \([a,b)\) are called half-open and refer to an interval with one point (the one next to the square bracket) included and one excluded.
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Set-builder notation. 🔗: To define a subset of some set \(S\text{,}\) we write \(\{ x \in S : \cdots \}\) where instead of the three dots we write a property of \(x\) that can be true or false. In some books the colon is replaced by a vertical line, the words "such that" or the abbreviation \(s.t.\) . If the set \(S\) is the set of all real numbers it is sometimes omitted. Thus, for example, \(\{ x : a \leq x \lt b \}\) refers to the half open interval of real numbers, \([a,b)\text{.}\)
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The defining colon-equal sign. 🔗: We use \(:=\) to mean that the quantity on the left is defined to be the quantity on the right, and a regular equal sign to mean an equation that could hold for some values of the variables and fail for others. Thus, \(f(x) := x^2 + 2\) defines a function, whereas \(f(x) = x^2 + 2\) is an equation which is true when a given function \(f\text{,}\) evaluated at \(x\text{,}\) has the same value as \(x^2 + 2\text{,}\) and false otherwise.

Checkpoint 7.

Suppose \(f(x) := x^2 + 2\text{.}\) What values of \(x\) make the equation

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\begin{equation*} f(x) = 5 - 3 x^2 \end{equation*}

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true? Please simplify your answer(s).

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The range of a function \(f\) is defined to be the set of all possible function values. Formally we can write the range of \(f\) as the set \(\{ f(x) : x \in \text{domain of } f \}\text{.}\)

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One final remark about the basic definitions: there is an ambiguity in common usage of the word "range". Sometimes "range" is used to refer to a bigger set than in our definition, namely the set of all things of the type that the function outputs (we’ll call this other set the target of the function, should we ever need to refer to it.). For example, they might say that the domain and range of a function \(f(x) := x^2 + 2\) is all real numbers. It’s fine to define the domain to be all real numbers, but then technically the range is the set of real numbers that are at least 2. If for some reason we want to tell the person we’re talking to that we intend the outputs to always be real numbers, we can say "the target is the real numbers".

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Checkpoint 8.

What are two formal mathematical ways of writing the set of real numbers that are at least 2, one using set notation and one using interval notation?

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Definition by cases.

As we said, the most familiar way of referring to a function is as a rule for converting input to output. Usually the rule is an equation, such as \(f(x) := C - x \cdot e^x\text{,}\) but the rule could be verbal, for example, "Let \(f(t)\) be the amount in tons of carbon dioxide emitted in \(t\) years." Sometimes we want to talk about functions that are defined by equations, but different ones in different parts of the domain. This is called definition by cases. An example from a recent research paper looks like this:

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\begin{equation*} f(x) := \begin{cases} -9x \amp a \leq -3 \\ 2x^2 - 3x \amp -3 \lt x \lt 1 \\ -a^3 \amp a \geq 1 \end{cases} \qquad . \end{equation*}

A number of useful functions can be defined in this way. For example the absolute value of \(x\text{,}\) denoted \(\left\lvert x\right\rvert\text{,}\) may also be defined in cases:

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\begin{equation*} |x| := \begin{cases} x \amp x \geq 0 \\ -x \amp x \lt 0 \end{cases} \qquad . \end{equation*}

Remark 0.4. Some remarks on defining by cases.

Note that \(x\) and \(-x\) agree at \(x=0\text{,}\) so we could have grouped zero with either case. When this happens, writing
\begin{equation*} |x| := \begin{cases} x \amp x \geq 0 \\ -x \amp x \leq 0 \end{cases} \qquad \end{equation*}
emphasizes this. If \(x\) and \(-x\) did not agree at \(x=0\text{,}\) this would be a badly formed definition.

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There is a period following the two example definitions but not the one in the first remark. Why? Because well written math follows rules of basic grammar. These rules can be a little different on occasion, but for the most part, you should expect this text to read in complete sentences, to define variables and functions before using them, and when used within sentences, to connect and flow logically, using connecting words like "and", "because", "therefore", and punctuation such as commas and periods.

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Checkpoint 9.

Which of the following defines a function whose domain is all real numbers?

\(\displaystyle f(x) := \begin{cases} x+1 \amp x \gt 2 \\ x-1 \amp x \lt 2 \end{cases}\)
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\(\displaystyle g(x) := \begin{cases} x+1 \amp x \geq 2 \\ x-1 \amp x \lt 2 \end{cases}\)
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\(\displaystyle h(x) := \begin{cases} x+1 \amp x \geq 2 \\ x-1 \amp x \leq 2 \end{cases}\)
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Answer.

\(\text{Choice 2}\)

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Free and bound variables.

In the defining statement \(f(x) := x^2 + 2\text{,}\) it would define the same function if instead we said \(f(u) := u^2 + 2\text{.}\) It is the same set of order pairs, has the same graph, etc. The variable \(x\) (or in the second case, \(u\)) is said to be a bound variable. The bound variable in this case runs over all values in the domain of \(f\text{.}\) A variable that is not bound is free. For example, in the definition \(f(u) := u^2 + c\text{,}\) the variable \(c\) is free. The definition of the function \(f\) depends on the value of \(c\text{.}\) If \(c = 2\text{,}\) it boils down to the previous definition. If \(c=1\) it is a different function. If \(c\) has not been assigned a value, then \(f\) is a function whose range is not the real numbers but rather algebraic expressions in the variable \(c\text{.}\)

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Bound variables arise many times throughout this course, in fact throughout math and throughout life! Here is a list of some places bound variables occur in this course, the first two of which you have already seen.

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In the definition of a function

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In the definition of a subset

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In limits

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In the definition of a derivative

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In summations

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In the definition of an integral

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In notions of orders of magnitude and asymptotic equivalence

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In Taylor’s theorem

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Let’s get some practice with identifying free and bound variables.

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Definition 0.5.

A function \(f\) is said to be differentiable at \(x\) if \(x\) is in the domain of \(f\) and \(f'(x)\) exists.

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Definition 0.6.

A function \(f\) is said to be differentiable on the open interval \((a,b)\) if \((a,b)\) is in the domain of \(f\) and if, for all \(x \in (a,b)\text{,}\) the derivative \(f'(x)\) exists.

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