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Function notation is a way to write functions that makes these mathematical relationships clear and easy to understand. Instead of saying "y equals something in terms of x," we use a special format that involves writing functions as \(f(x)\). Here, \(f\) signifies the function, and \(x\) is the variable. This notation provides a compact form that is handy in algebra and calculus. In the exercise, the original function is written as \(f(x) = 3x - 4\). It tells us that the output, denoted by \(f(x)\), depends on the given input \(x\) according to the rule "multiply \(x\) by 3 and then subtract 4." This notation is crucial when talking about inverse functions, as it helps keep track of which function we are dealing with, and when using operations, this clarity prevents errors.

Algebraic manipulation involves using algebraic techniques to rearrange equations or expressions in a more convenient form. In our exercise, we moved through a series of steps to simplify the function and solve for the variable \(y\). Let's break it down:

• **Substitution**: First, we substituted \(f(x)\) with \(y\), making the equation more straightforward as \(y = 3x - 4\).
• **Swapping variables**: We switched \(x\) and \(y\), turning our focus towards finding \(y\) as a function of \(x\) for the inverse.
• **Adding/Subtracting**: We used addition to move constants across the equation. By adding 4 to both sides of \(x = 3y - 4\), we kept the equation balanced.
• **Division**: Lastly, we divided everything by 3. This step is essential because it isolates \(y\), giving us our inverse function \(y = \frac{x + 4}{3}\).

These manipulations are basic yet powerful tools allowing us to change the function's form and uncover its inverse.

When we solve equations, especially when finding inverse functions, we essentially unravel the original function. We aim to "reverse" the operations. Here’s how we tackled it:To find an inverse function, the operations done by the original function must be undone in reverse order. Our original function \(f(x) = 3x - 4\) can be read as "multiply by 3 and subtract 4." For the inverse, we must add 4 and then divide by 3 in reverse.

• **Step 1**: The equation after swapping \(x\) and \(y\) was \(x = 3y - 4\). Adding 4 to both sides (undoing subtraction) leads to \(x + 4 = 3y\).
• **Step 2**: To reverse multiplication by 3, divide the whole equation by 3, yielding \(y = \frac{x + 4}{3}\).

The output, or inverse function, \(f^{-1}(x) = \frac{x + 4}{3}\), thus gives us the means to switch between inputs and outputs from our original function using each other's operations. By solving, inverse functions allow us to work backward, making them valuable for numerous applications in mathematics.