91Ó°ÊÓ

In mathematics, a function can be thought of as a machine that takes an input and provides an output based on a pre-defined rule. In our exercise, the function is given by __\(f(x) = 2x\)__. Here, the function simply multiplies the input, \(x\), by 2 to produce the output. This is known as function multiplication.
In practice, for every value of \(x\) you supply to the function \(f\), you get twice that value as output. For instance, if \(x = 3\), then \(f(3) = 2 * 3 = 6\). Similarly, for \(x = 5\), \(f(5) = 2 * 5 = 10\). These examples demonstrate how the function follows its rule to multiply the input by 2.

An inverse function is essentially the reverse of the original function. If a function \(f\) takes an input \(x\) and produces an output \(y\), its inverse function \(f^{-1}\) will take \(y\) as input and produce the original \(x\) as output. For the given function \( f(x) = 2x \), we are tasked with finding its inverse.
To unravel the definition of \(f^{-1}\), we start by setting __\(y = f(x)\)__. For our function, this translates to \(y = 2x\). The inverse function \(f^{-1}(y)\) should satisfy the property that applying it to \(y\) gives us back \(x\). By solving \(y = 2x\) for \(x\), we isolate \(x\) as \(x = \frac{y}{2}\), hence the inverse function \(f^{-1}(y)\) is __\(\frac{y}{2}\)__. Replacing \(y\) with \(x\) yields the inverse function \(f^{-1}(x) = \frac{x}{2}\).

Algebraic manipulation is a process used to simplify and solve equations. In finding the inverse function, we rely heavily on these algebraic techniques. Starting from the function \(f(x) = 2x\), recognizing the need for its inverse means we start with \(y = 2x\).
We want to solve this equation for \(x\). Using basic algebra, we divide both sides of the equation by 2, giving us \(x = \frac{y}{2}\). This step is essential as it rearranges the equation to isolate \(x\) in terms of \(y\). In doing so, we determine that \(f^{-1}(x) = \frac{x}{2}\). This process of manipulating the equation is critical to understanding and finding inverse functions.

Every function can be seen as a unique input-output relationship. For the function \(f(x) = 2x\), the relationship is clear: input \(x\), multiply by 2, and get an output. The inverse function reverses this relationship.
For the given function \(f(x) = 2x\), if you input a value, say \(x = 4\), the output will be \(2 * 4 = 8\). The inverse function works backward. Given the output 8, and knowing that \(f^{-1}(x) = \frac{x}{2}\), you divide by 2 to retrieve the original input: \(\frac{8}{2} = 4\). This reciprocal relationship forms the basis of understanding how functions and their inverses interact and ensure they map correctly back and forth between inputs and outputs.