Determine the inverse of the given equation: \( y=\sqrt{x-4}+3 \)

To find the inverse of the equation \( y = \sqrt{x - 4} + 3 \), start by swapping \( x \) and \( y \) in the equation to get \( x = \sqrt{y - 4} + 3 \). Then, isolate \( y \) by subtracting 3 from both sides: \[ x - 3 = \sqrt{y - 4} \] Next, square both sides to eliminate the square root: \[ (x - 3)^2 = y - 4 \] Finally, solve for \( y \) by adding 4: \[ y = (x - 3)^2 + 4 \] So, the inverse function is \( y = (x - 3)^2 + 4 \). This means if you input \( x \) into this new equation, it will give you the original \( y \) value from the first equation! Exploring the world of function inverses opens up a treasure trove of possibilities! These functions can tell us about symmetry, meaning the original function and its inverse reflect across the line \( y = x \). When graphed, they mirror each other, showcasing a beautiful mathematical relationship. If you’re ready to delve deeper, why not check out some fun applications of inverses? They're crucial in fields like cryptography, where encoding and decoding messages often involves reversing processes. Mathematicians and computer scientists alike rely on these concepts to keep information secure and accessible!