Find the inverse of a one-to-one function.

Understand the Problem

The question is asking how to determine the inverse of a one-to-one function, which involves finding a function that reverses the effect of the original function. To solve this, we generally swap the input and output of the function and solve for the new output in terms of the new input.

Answer

The inverse function is $f^{-1}(x)$.

Steps to Solve

Identify the original function Begin with the original one-to-one function, which we can represent as $y = f(x)$.
Swap variables To find the inverse, swap the variables $x$ and $y$. This gives us the equation $x = f(y)$.
Solve for $y$ Now, solve the equation for $y$. This may involve algebraic manipulations depending on the form of the original function.
Express the inverse function Once you have $y$ isolated, express the inverse function as $f^{-1}(x)$, indicating that this function reverses the effect of $f$.
Verify the inverse Finally, to confirm that you have found the correct inverse, check if $f(f^{-1}(x))$ and $f^{-1}(f(x))$ both simplify to $x$.

The inverse function is represented as $f^{-1}(x)$, where $x$ is obtained through the steps outlined depending on the original function.

More Information

Finding the inverse of a one-to-one function is essential in various fields, including mathematics, engineering, and physics. The concept is crucial for solving equations that can be expressed in different forms, leading to increased understanding of function relationships.

Tips

Not ensuring the function is one-to-one: Only one-to-one functions have inverses.
Forgetting to swap variables and solve correctly: Ensure you accurately swap and manipulate the equation to isolate $y$.
Failing to verify the inverse function: Always confirm by plugging the inverse back into the original function.

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