91Ó°ÊÓ

In mathematics, functions are a core concept, and learning how to express and manipulate them effectively is crucial. Function notation is a way of representing functions that makes it easier to understand and work with them. In our exercise, we start with a function given as: \(f(x) = \frac{1}{x+2}\)Here, \(f\) represents the function, and \(x\) is the variable. The expression \(f(x)\) is read as 'f of x'. Using \(f(x)\) helps us denote that this expression is a function of \(x\). This notation is powerful as it allows us to see what the function does to the input \(x\). It's quite like a machine where you input something and it outputs something else. The notation makes it clear what happens to the input.

Algebraic manipulation involves rearranging and simplifying expressions. It is essential in solving for variables and finding inverse functions. In our steps, we replaced \(f(x)\) with \(y\) to simplify our work: \(y = \frac{1}{x+2}\)Next, we swapped \(x\) and \(y\) to prepare for finding the inverse: \(x = \frac{1}{y+2}\)With this swap, our goal is to solve for \(y\) again. First, by multiplying both sides by \(y + 2\): \[ x(y + 2) = 1 \]Then, distribute \(x\) to both terms inside the parenthesis: \[ xy + 2x = 1 \]To isolate \(y\), we subtract \(2x\) from both sides: \[ xy = 1 - 2x \]Finally, we solve for \(y\) by dividing both sides by \(x\): \[ y = \frac{1 - 2x}{x} \]This method shows how algebraic manipulation is used to isolate and solve for a variable.

Solving equations is the process of finding unknown values that satisfy given equations. It's essential for solving for inverse functions. In our steps, we solved the equation to find \(y\), which represents the inverse function. After swapping \(x\) and \(y\), and performing algebraic manipulations, we arrived at: \[ y = \frac{1 - 2x}{x} \]This equation now represents the inverse function. The final step is substituting \(y\) with \(f^{-1}(x)\): \[ f^{-1}(x) = \frac{1 - 2x}{x} \]This notation \(f^{-1}(x)\) indicates that it is the inverse function of the original. Solving equations involves systematically applying algebraic rules and operations to find the desired variable. Remember, every step you take must follow logical and mathematical principles, ensuring that the equations remain balanced and true.