How to Find the Inverse of a Function

Hey there, awesome students! 🎉 Are you excited to learn about inverse functions? Let’s dig in and uncover how to find the inverse of a function! Once you understand this, you'll feel like a math genius! 🧙‍♂️✨ So, let's get started!

Step 1: What is an Inverse Function?

First, let's talk about what an inverse function is. An inverse function "reverses" what the original function does.

If you have a function called $f(x)$ that turns $x$ into $y$, the inverse function, written as $f^{-1}(y)$, turns $y$ back into $x$.

This means if you take $x$, apply $f$, and then apply $f^{-1}$ to the result, you'll get back to your original $x$. We can show this with:

$f(f^{-1}(y)) = y$

and

$f^{-1}(f(x)) = x$

So, the output of the original function becomes the input for the inverse function!

Step 2: Check if a Function has an Inverse

Not all functions have inverses! 🎭 To find out if a function has an inverse, we use something called the Horizontal Line Test. Here’s how to do that:

• Graph the function: If you can draw it, go ahead and graph the function.
• Draw horizontal lines: Imagine drawing horizontal lines across the graph. If any horizontal line crosses the graph more than once, then the function does not have an inverse.
• Conclusion: Functions that pass this test (meaning every horizontal line crosses the graph only once) are one-to-one and have inverses!

Step 3: Write the Function as an Equation

Let’s say we have a function like $f(x) = 2x + 3$. To find the inverse, we start by writing it out:

1. Write the function:
   $y = f(x) = 2x + 3$

Step 4: Switch x and y

Now comes a fun step—switching the variables! 🎊 This is crucial for finding the inverse.

• Switch $x$ and $y$:
  $x = 2y + 3$

Step 5: Solve for y

Now we want to solve for $y$ to find the inverse function—let’s go for it!

1. Subtract 3 from both sides:
   $x - 3 = 2y$

2. Divide by 2:
   $y = \frac{x - 3}{2}$

Step 6: Write the Inverse Function

Yay! 🎉 We’ve got it! The inverse function is:

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

If you plug the output of the original function into this inverse function, you’ll get back to your original input!

Step 7: Double-Check Your Work

To make sure we did everything right, let’s check both functions:

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

2. Find $f^{-1}(f(x))$:
   $f^{-1}(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x$

Both checks show we found the right inverse! 🎉🤩

Now it's your time to shine! Go tackle those inverse functions with confidence! You've got this! 🚀