Determine the algebraic inverse function by swapping x and y and solving for y | Step-by-Step Solution

Finding the Inverse Function

What We're Solving:

We need to find the inverse of y = ln(ln(x-3)) + 1. This means we want to find a function that "undoes" what our original function does to any input value.

The Approach:

To find an inverse function, we use the "swap and solve" method:

• 1. Swap x and y in our equation
• 2. Solve the new equation for y
• 3. The result is our inverse function f⁻¹(x)

If our original function takes some input and transforms it through a series of steps, the inverse function does those same steps backwards to get us back to where we started!

Step-by-Step Solution:

Step 1: Write our original equation y = ln(ln(x-3)) + 1

Step 2: Swap x and y x = ln(ln(y-3)) + 1

Now we need to solve for y. We'll "peel away" each operation, working from the outside in!

Step 3: Isolate the first natural logarithm x - 1 = ln(ln(y-3)) (We subtracted 1 from both sides)

Step 4: Use the exponential function to "undo" the outer ln e^(x-1) = ln(y-3) (Since e^(ln(something)) = something)

Step 5: Use the exponential function again to "undo" the inner ln e^(e^(x-1)) = y-3 (Same principle: e^(ln(something)) = something)

Step 6: Solve for y y = e^(e^(x-1)) + 3 (We added 3 to both sides)

The Answer:

The inverse function is: f⁻¹(x) = e^(e^(x-1)) + 3

Memory Tip:

Think of inverse functions like getting dressed and undressed! If you put on socks, then shoes, then a coat, you'd take them off in reverse order: coat, shoes, then socks.

Our original function: subtract 3, take ln, take ln again, then add 1 Our inverse function: subtract 1, take e^x, take e^x again, then add 3

The operations are reversed AND in opposite order! 🧦👟🧥

Great job working through this challenging composite function! The key is patience and working step-by-step to "undo" each operation.