91Ó°ÊÓ

In mathematics, function notation is a way to express relationships between variables. We use this notation to clearly define functions, like stating that \( f(x) = x + 5 \), where \( f \) tells us which function to apply to the variable \( x \). When working with function notation, the goal is often to find the inverse of a function. An inverse function essentially "reverses" the effect of the original function.
When you have an equation like \( y = f(x) \), the inverse will be the function that returns \( x \) when given \( y \). The notation \( f^{-1}(x) \) symbolizes this inverse function. Understanding and finding the inverse is important as it allows us to see what input produces a specific output in original scenarios. In this exercise, we're tasked with finding the inverse of \( f(x) = x + 5 \).

To uncover the inverse of a function like \( f(x) = x + 5 \), we need to master solving equations. The primary step is to express the function as \( y = f(x) \), which gives us \( y = x + 5 \). The goal is to solve this equation for \( x \). Start by isolating \( x \) on one side of the equation. Here, you take the equation \( y = x + 5 \) and subtract 5 from both sides, resulting in \( x = y - 5 \).
This step is crucial because it reverses the manipulation applied by the original function. By isolating \( x \), you’ve expressed it in terms of \( y \), essentially setting the stage to rewrite it as the inverse function. Once \( x \, \) is isolated, you can proceed to swap the variables.

Algebraic manipulation involves the process of rearranging and simplifying expressions. In the case of finding an inverse function, this often involves swapping the variables \( x \) and \( y \) from the expression we derived by solving for \( x \). After obtaining \( x = y - 5 \), we swap \( x \) and \( y \) to reflect the inverse relationship, leading to \( y = x - 5 \).

• Swapping the variables signifies that for every \( y \) you input, you'll get an \( x \) in return, reversing the original function's mapping.
• The final expression \( f^{-1}(x) = x - 5 \) stands for the inverse function, meaning if \( x \) gets increased by 5 in the original function, the inverse will decrease it by 5.

The swap and rewrite complete the process, showing a clear, logical method to derive an inverse function from an original function using algebraic manipulation.