Determine function for table of values

Question

Problem:

Given that a function $ f $ is completely defined by the following table: $$ \begin{array}{c|lcr} \\ \hline x & 0 & 2 & 3 \\ f(x) & -2 & 4 & √10 \\ \end{array} $$ Write down the function. State the rule that determines the function. What is the domain of definition. What is the range. What is the meaning of f(0)? What is the meaning of f(1)?

Background:

The context is an introductory chapter on functions. The author gives a few examples of how given a set X and a set Y, a rule is given which determines a unique element y of Y whenever an element x of X is selected. The rule generates the third set, namely the set of ordered pairs (x,y)

The author gives a few examples of rules.

1. Postage is 6 cents per ounce
2. Gives a table of logarithm values (10,1.0000), (25,1.3979) (50,1.6990) ..
3. b = a^3 for a = -3, -2, -1, 0, 1, 2, 3

He then proceeds to define a function as:

1. a set X called the domain
2. a set Y called the range
3. a rule which assigns to each element of X a unique element of Y and which assigns each element of Y to one or more elements of X.

"or more compactly...

a function f is a set of ordered pairs (x,y) where (1) x is an element of a set X, (2) y is an element of a set Y, and (3) no two pairs in f have the same first element. And that therefore a function is a kind of a subset of the cartesian product X times Y."

He then provides a table which lists the function f for each example, the rule determining f, the domain and the range. He highlights the difference between the case:

where the rule is a table of values such as example 2 above (logarithms) pointing out that the table is only a portion, and that a formula was used to derive the entries in the table. Therefore the formula should be considered as the rule, rather than the table itself.

with

another case where he lists a finite table of values $$ \begin{array}{c|lcr} \\ \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ f(x) & 0 & 27 & -8 & 1 & -27 & 8 & -1\\ \end{array} $$and states that the function is completely defined by the table.

Original Question:

I've come up with $ f( x) = 3x - 2$ which fits the first pair of values but cannot determine relationship for $(3,\sqrt{10})$ and since the question says $f$ is completely defined then I'm not sure how to proceed.

I understood the author to mean "completely defined by" meaning that there is one formula or rule which translates the x values to the y values in the table. And I'm not sure what that formula might be. Also not sure what he means by what is meaning of f(0) and f(1).

Answer 1

The context is an introductory chapter on functions. The author gives a few examples of how given a set $X$ and a set $Y$, a rule is given which determines a unique element $y$ of $Y$ whenever an element $x$ of $X$ is selected.

Ah this helps. In my original comment, I assumed the domain of $f$ is $\mathbb{R}$ (As with most questions of this level on this site). But given this is from an introduction of functions, it makes sense that you should give the domain yourself (And maybe also the codomain).

Well, if $f$ is uniquely determined by this table, then the only possibility is that the domain of $f$ is $\{0,2,3\}$: If there is some other $x$ in the domain of $f$, say the domain is $\{0,1,2,3\}$, we could choose different values for $f(1)$ and get different functions.

So the function should be $f\colon\{0,2,3\}\to\{-2,4,\sqrt{10}\}$ given by $f(0)=-2$ and $f(2)=4$ and $f(3)=\sqrt{10}$, as Paul said.