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Verifying a function and its inverse is an essential step in understanding inverse functions. This process confirms that the inverse relationship truly holds. To verify, we take two specific paths:

• We substitute the inverse function into the original function and check if the result simplifies back to the input of the inverse function. This is denoted as checking whether \(f(f^{-1}(x)) = x\).
• We substitute the original function into the inverse function and simplify to see if we obtain the input of the original function. This is represented by verifying \(f^{-1}(f(x)) = x\).

For the function \(f(x) = 3x + 1\), its inverse \(f^{-1}(x) = \frac{x-1}{3}\) perfectly satisfies both conditions. This means that when we apply the inverse function, it effectively "undoes" the operation of the original function, bringing us back to where we started. This mutual cancellation property is a key characteristic of inverse functions.

A linear function is a type of function which creates a straight line when graphed. It usually follows the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These functions are of fundamental importance in algebra as they establish a simple, constant rate of change.
In our example, the function \(f(x) = 3x + 1\) is a linear function. Here, \(3\) represents the slope, indicating the function increases by 3 units vertically for every unit increase horizontally. The \(+1\) represents the y-intercept, showing that the line crosses the y-axis at 1.
Understanding linear functions is crucial because they model real-world relationships where one quantity depends on another in a linear manner. For example, linear functions can describe relationships like speed over time or currency exchange rates.

Solving equations is a core skill in algebra, where we aim to find the value of the variable that satisfies the equation. It involves manipulating the equation to isolate the variable on one side.
In the context of finding an inverse function, solving equations allows us to determine the inverse specifically. Starting with the equation \(y = 3x + 1\), we reverse the roles of \(x\) and \(y\), giving us \(x = 3y + 1\). To solve for \(y\), we:

• Subtract 1 from both sides, resulting in \(x - 1 = 3y\).
• Divide both sides by 3, yielding \(y = \frac{x-1}{3}\).

This process shows how solving equations can transform a given function into its inverse. Mastering this skill is valuable across various mathematical problems and disciplines.