The inverse of a function is a function that "reverses" the effect of the original function. Specifically, if $f(x) = y$, then the inverse function, denoted by $f^{-1}(y)$, satisfies $f^{-1}(y) = x$. This means that applying the inverse function to the output of the original function will return the input.

Requirements for a function to have an inverse:

1. One-to-One (Injective): The function must map each input to a unique output (no two different inputs can map to the same output). This ensures that the inverse function can correctly "reverse" the mapping.
2. Onto (Surjective): Every element in the range of the function must be the image of some element in the domain, ensuring that the inverse covers the entire domain of the original function.

For the inverse function to exist, the original function $f$ must be bijective (both injective and surjective). If a function is not one-to-one, an inverse does not exist because multiple inputs could correspond to the same output, making it impossible to reverse the function uniquely.

Steps to find the inverse of a one-to-one function:

1. Express the function in terms of $y$: Start with the original function written as $y = f(x)$.

2. Interchange $x$ and $y$: Swap $x$ and $y$ in the equation. This reflects the idea that for the inverse, what was once the input ($x$) is now the output, and what was once the output ($y$) is now the input.

3. Solve for $y$: Solve the new equation for $y$, which represents $f^{-1}(x)$, the inverse function.

Example:

Let’s consider the example function $f(x) = 4x + 6$.

Step 1: Write the function in terms of $y$

$y = 4x + 6$

Step 2: Interchange $x$ and $y$

$x = 4y + 6$

Step 3: Solve for $y$

Subtract 6 from both sides: $x - 6 = 4y$

Now, divide both sides by 4: $y = \frac{x - 6}{4}$

Thus, the inverse function is: $f^{-1}(x) = \frac{x - 6}{4}$

Key Point:

The inverse function of $f(x) = 4x + 6$ is $f^{-1}(x) = \frac{x - 6}{4}$.

A Note on Many-to-One Functions:

If the original function is not one-to-one (i.e., it’s many-to-one), it cannot have an inverse function in the traditional sense. However, in some cases, we can restrict the domain of the original function to make it one-to-one and then find the inverse.

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Do you want further details on any step, or have any questions about this process?

Here are five follow-up questions for you to consider:

1. What is the inverse of the function $f(x) = 3x^2$ when restricted to $x \geq 0$?
2. How can you tell if a function is one-to-one by looking at its graph?
3. What happens to the domain and range when finding the inverse of a function?
4. Can a non-injective function have an inverse if we restrict its domain?
5. Why is the horizontal line test useful when determining if a function has an inverse?

Tip: Always verify the inverse by checking if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. This ensures the correctness of your inverse function.