Appendix A. Are Variables Dummies?
There are two schools of thought when using variables in mathematics.
Dummy Variables.
Input variables are used merely as placeholders in function definitions. Any name for a variable is as good as any other. We prefer to use function names over variables when possible. Output variables are largely unnecessary.
Meaningful Variables.
A variable should be given meaning when possible. Choosing good variable names helps simplify notation. We should only reuse a variable name when that variable refers to the same value: it is a mistake to have both \( x = 2 \) and \( x = 5 \) at the same time in the same context.
Both schools of thought on this matter are perfectly valid! That said, the author tends to prefer using meaningful variables. Calculus is already difficult enough, and so we should be on the lookout for ways to be more easily understood. Let's take a look at how we compose functions using meaningful variables, and how this works using dummy variables.
Example.
Consider the functions
\[
\begin{align}
z &\mathrel{\overset{f}{\leftarrow}} u
&
&
&
u &\mathrel{\overset{g}{\leftarrow}} x
\\
z &= 3 u
&
&\andSpaced
&
u &= x^2 + 1.
\end{align}
\]
Let see how \( z \) depends on \( x \) by making a substitution.
\[
\begin{align}
z &= 3 \mult \bigl(x^2 + 1\bigr)
\end{align}
\]
We'll call \( u \) an intermediate variable because it is an input variable for \( f \) and an output variable for \( g \).
In the previous example our intermediate variable, \( u \), is a meaningful variable: reusing the variable name allows us to infer what substitution to make. Let's redo the example using dummy variables to see how that style works.
Example.
Consider the functions
\[
\begin{align}
f \of x &= 3 x
&
&\andSpaced
&
g \of x &= x^2 + 1
\end{align}
\]
where we regard both uses of \( x \) as dummy variables. Let's find \( f \of g \of x \) by substituting.
\[
\begin{align}
f \of g \of x &= f \of \bigl(x^2 + 1\bigr) = 3 \mult \bigl(x^2 + 1\bigr)
\end{align}
\]
This works, but it would be very easy to mix something up. Let's see why by looking at the schematic for this composite.
We've used the same variable name \( x \) for both the input and the output of \( g \). In situations like this, we'll prefer to choose distinct variable names to help reduce confusion.