A function can be defined via a graph. Each dot on a graph is highlighting a point and the coordinates of this point define a pair in the function. We can use a graph to estimate function values, but we cannot escape the approximation inherent to drawing. To communicate about exactness, some functions have an algebraic description of the pairings. We call this algebraic tool a formula or an equation. Not all functions have formulas with which we can calculate. But when they do, there is an operation manual to follow.

$$\begin {array}{rcl} \text {expression involving domain number} & = & \text {function value at the domain number} \\ \text {expression involving domain number,} \, d & = & \text {function,} \, f \text {, value at the domain number,} \, d \\ \text {expression involving domain number,} \, d & = & f(d) \end {array}$$

Operation Manual

A formula is a tool. We use it to connect domain numbers to their range partners. And, like any tool, it has an Operation Manual.

The formula in the example above is $3k - 2$. The equation $P(k) = 3k - 2$ tells us this formula calculates range values for the function $P$. How would we use this formula to calculate the value of $P$ at $5$? In other words, how would we use it to calculate $P(5)$?

We just gave ourselves our first clue. We went from $P(k)$ to $P(5)$ by replacing $k$ with $5$. We should do the samething with the formula. However, this doesn’t always work.

Replacing $k$ with $5$ in $3k - 2$ gives us $35-2$, which equals $33$ and $33$ is not $P(5)$. The problem is that our formula is using shorthand notation. Simply replacing the variable with the domain number fails to maintain the meaning of the expression. In this case, a number next to a variable is shorthand for multiplication and this was lost when we replaced $k$ with the $5$.

We want to replace all occurrences of the variable with the domain number, while maintain the meaning of the expression. As you gain experience with formulas, you will be able to do this on-the-fly. But a quick rule-of-thumb that cures this problem is to first replace all occurrences of the variable in the formula with the variable wrapped in parentheses.

$$P(k) = 3(k) - 2$$

Once you are comfortable with the meaning of a formula, then you can see the parentheses in your head and no longer need to write them.

$\blacktriangleright$ DNE or dne stands for Does Not Exist.

A formula describes the connection between domain numbers and their range partners. Given a domain number, substitution, and then direct calculation gives the function value at that domain number. We can also go the other way. Given a range value or function value, we can find the domain numbers connected to it.

We do this by creating an equation involving function notaion.

There is still a question of whether or not we have all of the domain numbers partnered with $-1$.

Two Questions

Questions about functions generally come in two types.

(a)

You know the domain number and you want the range partner or function value. This question is answered with evaluation. Mostly you evaluate a formula or you track the function value down inside a graph.

(b)

You know the function value and you want the associated domain values. This question is usually phrased in terms of an equation to solve. The solution set may include more than one domain number.