To find the inverse of a function, you can follow some simple steps. Let’s break it down together!

Step 1: Understand the Function

First, check if the function is one-to-one. This means that it passes the horizontal line test. In simple terms, this means that different x-values give you different y-values.

For example:
The function ( f(x) = x^2 ) is not one-to-one because both ( f(2) ) and ( f(-2) ) give you the same value (4).

But the function ( f(x) = 2x + 3 ) is one-to-one.

Step 2: Replace ( f(x) ) with ( y )

Once you see that the function is one-to-one, rewrite the function as ( y = f(x) ).

So if your function is ( f(x) = 2x + 3 ), write it like this:
$y = 2x + 3$

Step 3: Swap x and y

Now, swap x and y in the equation. This is like flipping the function over the line ( y = x ).

Using our example, it becomes:
$x = 2y + 3$

Step 4: Solve for y

Next, solve the equation for y. This means you want to get y all by itself.

For our example:

1. Subtract 3 from both sides:
   $x - 3 = 2y$
2. Then divide by 2:
   $y = \frac{x - 3}{2}$

Step 5: Write the Inverse Function

Now that you have y by itself, write it using inverse notation, which looks like this: ( f^{-1}(x) ).

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

Step 6: Verify the Inverse

Finally, check if your inverse function is correct by testing it with the original function. You should get back to your starting value:

• First, calculate ( f(f^{-1}(x)) ) and see if it equals ( x ).
• Then try ( f^{-1}(f(x)) ) and check if it also equals ( x ).

Let’s see how that works with our function:

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

And that’s it! You’ve successfully found the inverse of the function.

Remember, practice makes perfect! So keep trying different functions to help you get better!