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Understanding what a one-to-one function is, begins with recognizing that such functions map each output to only one input. This is a key property because if a function is not one-to-one, it cannot have an inverse. To visualize this, imagine a function where each input value (x) gives a unique output (f(x)). If you have distinct inputs, they correspond to distinct outputs. This relationship ensures that by knowing the output, you can determine a unique input. A quick test to check if a function is one-to-one is the Horizontal Line Test. Draw a horizontal line across the graph of the function. If at any point the line crosses more than one part of the graph, the function is not one-to-one. In the context of the given function, verifying one-to-one ensures an inverse exists, which is vital for tasks like finding an inverse function. Here, the function given is stated to be one-to-one, enabling us to proceed with finding its inverse confidently.

Algebraic manipulation is a crucial process in solving for inverse functions. It involves reshaping and simplifying expressions to isolate desired variables. Let's break down the process using our function, where we started by setting it as an equation with y representing the function itself:

• Initially, we had: \( y = \frac{3 - x}{2x + 1} \).
• Then, we swapped the roles of \( x \) and \( y \), which transformed the equation to \( x = \frac{3 - y}{2y + 1} \).
• To eliminate the fraction, we multiplied both sides by \( 2y + 1 \), resulting in \( x(2y + 1) = 3 - y \).
• We rearranged terms to isolate \( y \), giving us \( 2xy + y = 3 - x \).
• By factoring out \( y \), we deduced \( y(2x + 1) = 3 - x \).
• Finally, we solved for \( y \) by dividing both sides by \( 2x + 1 \), achieving \( y = \frac{3 - x}{2x + 1} \).

Each algebraic step was necessary to achieve our final goal: to express \( y \) independently of any equation conditions, revealing that the inverse closely resembles the original function.

Once we find what appears to be an inverse function, verifying it ensures that the function truly reverses the original mapping. This involves substitution and simplification to see if the function returns the input.To verify, take the inverse function \( f^{-1}(x) \) and plug it back into the original function to see if we can resolve it to \( x \):

• Using \( f(f^{-1}(x)) \), substitute \( f^{-1}(x) = \frac{3 - x}{2x + 1} \) back into \( f(x) = \frac{3 - x}{2x + 1} \).
• Simplify to show it equates back to \( x \).

In this case, because the function is its own inverse, performing these steps verifies that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). This property highlights the essential characteristics of inverse functions. Ensuring this verification is a critical step as it validates all previous algebraic work and confirms the ability to reverse the function successfully.