To find the inverse of a function, you essentially reverse the roles of the input (independent variable) and output (dependent variable) and then solve for the new output variable. Here's a step-by-step guide:

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

Given a function $f(x)$, start by expressing it as: $y = f(x)$ This means that $y$ is the output when $x$ is the input.

Step 2: Swap $x$ and $y$

To find the inverse, switch the roles of $x$ and $y$. Replace $y$ with $x$ and $x$ with $y$: $x = f(y)$

Step 3: Solve for $y$

Now, solve this equation for $y$. The solution you obtain will be the expression for the inverse function, which we'll denote as $f^{-1}(x)$.

Step 4: Verify the Inverse

To ensure your inverse is correct, you can perform the following checks:

1. Check if $f(f^{-1}(x)) = x$: Substitute $f^{-1}(x)$ into the original function and see if you get back $x$.
2. Check if $f^{-1}(f(x)) = x$: Substitute $f(x)$ into the inverse function and see if you get back $x$.

Example

Let's find the inverse of the function $f(x) = 2x + 3$.

1. Express the function: $y = 2x + 3$

2. Swap $x$ and $y$: $x = 2y + 3$

3. Solve for $y$: $x - 3 = 2y$ $y = \frac{x - 3}{2}$

   So, the inverse function is: $f^{-1}(x) = \frac{x - 3}{2}$

4. Verify the inverse:

   • Check $f(f^{-1}(x))$: $f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x$
   • Check $f^{-1}(f(x))$: $f^{-1}(2x + 3) = \frac{2x + 3 - 3}{2} = \frac{2x}{2} = x$ Both checks confirm that the inverse is correct.

Would you like more details or have any questions?

Related Questions:

1. How do you find the inverse of a quadratic function?
2. Can every function have an inverse? Why or why not?
3. What is the geometric interpretation of the inverse function on a graph?
4. How can you find the inverse of a function with a square root?
5. How do you verify if two functions are inverses of each other?

Tip: Remember that a function must be one-to-one (each $x$ maps to exactly one $y$ and each $y$ maps to exactly one $x$) for its inverse to be a function.