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Understanding function inverses is crucial in mathematics. The concept involves swapping the input and output of a function. For a function \(f(x)\), its inverse \(f^{-1}(x)\) is found by reversing the process of \(f(x)\). In other words:

• The input of the original function becomes the output of the inverse function.
• The output of the original function becomes the input of the inverse function.

To find an inverse function, follow these steps:Replace \(f(x)\) with \(y\). Then, swap the variables \(x\) and \(y\). Rearrange the expression to solve for \(y\). After completing these steps, you've successfully found the inverse function!In our example, using \(f(x)=\frac{2}{3} x-3\), we first switch \(f(x)\) to \(y\), swap \(x\) and \(y\), and then solve for \(y\) to get the inverse: \(y=\frac{3x+9}{2}\). This new equation for \(y\) represents \(f^{-1}(x)\).

The horizontal line test is a simple way to determine if a function's inverse is also a function. A function is one-to-one (bijective) if it passes this test, meaning it has an inverse that is also a function. Here’s how it works:Imagine drawing horizontal lines across the graph of the function. If any horizontal line intersects the graph more than once, the function fails the test and does not have an inverse that is a function.In our example, the function \(f(x)=\frac{2}{3} x-3\) is a linear function, which always passes the horizontal line test. Any horizontal line drawn would intersect this linear graph only once, confirming that its inverse is indeed a function.This method ensures that the inverse relationship properly swaps the input and output, maintaining a one-to-one relationship.

Algebraic manipulation involves rearranging equations to solve for a specific variable. To find an inverse function, this skill is essential. Let's break down the process using the problem \(f(x)=\frac{2}{3} x-3\):

• Begin by expressing the function in terms of \(y\): \(y=\frac{2}{3}x-3\).
• Swap variables: \(x=\frac{2}{3}y-3\).
• Clear the fraction by multiplying through by 3: \(3x=2y-9\).
• Isolate \(y\): Add 9 to both sides, giving \(3x+9=2y\).
• Divide by 2 to solve for \(y\): \(y=\frac{3x+9}{2}\).

By applying algebraic manipulation, you transform the original function into its inverse form confidently, ensuring that each step maintains the integrity of the equation. This logical progression allows for a smooth transition from the original function to its inverse.