To find the inverse of a function, you need to follow these steps. The inverse of a function $f(x)$ is denoted by $f^{-1}(x)$. The idea is that if $y = f(x)$, then $x = f^{-1}(y)$.

Steps to Find the Inverse of a Function

1. Start with the function equation: Begin with the equation $y = f(x)$.

2. Swap the variables: Replace $y$ with $x$ and $x$ with $y$. This reflects the idea that in the inverse function, the roles of the input and output are reversed.

   $x = f(y)$

3. Solve for $y$: Manipulate the equation algebraically to solve for $y$ in terms of $x$. The expression you obtain will be the inverse function $f^{-1}(x)$.

4. Express the inverse function: Once you have solved for $y$, rewrite the equation as $y = f^{-1}(x)$.

5. Verify the inverse (optional): To ensure correctness, you can verify by checking that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Example:

Find the inverse of the function $f(x) = 2x + 3$.

1. Start with the function equation:

   $y = 2x + 3$

2. Swap the variables:

   $x = 2y + 3$

3. Solve for $y$:

   $x - 3 = 2y$

   $y = \frac{x - 3}{2}$

4. Express the inverse function:

   $f^{-1}(x) = \frac{x - 3}{2}$

Verification:

To verify, substitute $f^{-1}(x)$ into $f(x)$:

• $f(f^{-1}(x)) = f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x$
• $f^{-1}(f(x)) = f^{-1}(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x$

Both checks confirm that the inverse is correct.

Things to Note:

• Not all functions have inverses. A function must be one-to-one (bijective) to have an inverse, which means it passes both the horizontal line test and vertical line test.
• The inverse of a function, if it exists, will reverse the input-output relationship of the original function.

Would you like to try finding the inverse of another function or explore more about one-to-one functions?

Related Questions:

1. How do you determine if a function is one-to-one?
2. What is the significance of the horizontal line test in determining inverses?
3. Can all quadratic functions have inverses?
4. How do you find the inverse of a function involving square roots?
5. What is the relationship between the graph of a function and its inverse?

Tip: Always check that the original function is one-to-one before attempting to find its inverse, as not all functions have inverses that are also functions.