Suppose $f(x)=2 x-1$ and $g(x)=\frac{x+1}{a}$. Which value(s) of $a$ would make the composite functions commutative?
We are given two functions \(f(x) = 2x - 1\) and \(g(x) = \frac{x+1}{a}\). We want to find the value of \(a\) that makes these functions commute, which means \(f(g(x)) = g(f(x))\) for all \(x\). We must assume \(a eq 0\) since it appears in the denominator of \(g(x)\).
First, let us calculate \(f(g(x))\). Substituting \(g(x)\) into \(f(x)\):
\[f(g(x)) = f\left(\frac{x+1}{a}\right) = 2\left(\frac{x+1}{a}\right) - 1 = \frac{2(x+1)}{a} - 1 = \frac{2x+2-a}{a}.\]
Next, let us calculate \(g(f(x))\). Substituting \(f(x)\) into \(g(x)\):
\[g(f(x)) = g(2x-1) = \frac{(2x-1)+1}{a} = \frac{2x}{a}.\]
For the functions to be commutative, we set the two expressions equal for all \(x\):
\[\frac{2x+2-a}{a} = \frac{2x}{a}.\]
Since \(a
eq 0\), we can multiply both sides of the equation by \(a\) to obtain:
\[2x + 2 - a = 2x.\]
Subtracting \(2x\) from both sides of the equation gives:
\[2 - a = 0,\]
which simplifies to \(a = 2\).
We have found that for \(a=2\), the condition \(f(g(x)) = g(f(x))\) is satisfied for all \(x\). Thus, the functions are commutative when \(a = 2\). The answer corresponds to the second option. The final answer is \(\boxed{2}\).
This problem is a fundamental introduction to the composition of functions and the concept of commutativity. In computer science and algorithm design, understanding whether two functions commute can be crucial for optimizing operation sequences. For instance, if two operations on a data set commute, the order in which they are performed does not change the result, which allows for parallel processing or reorganization of steps to improve efficiency. This algebraic approach provides a framework for testing such properties in more complex systems.