Consider the function \( f(x) = \sqrt{x} \). Describe how you would graph its inverse, and explain any transformations that occur.

To graph the inverse of the function \( f(x) = \sqrt{x} \), start by recognizing that the inverse function will reflect across the line \( y = x \). First, we'll swap the x and y values, leading us to the inverse function \( f^{-1}(x) = x^2 \), valid for \( x \geq 0 \). To graph this, begin by sketching \( f(x) = \sqrt{x} \), which is a curve starting from the point (0,0) and rising to the right. Then, reflect all points across the line \( y = x \); you'll now have a parabola opening to the right, starting from (0,0) and curving upwards along the x-axis. As for transformations, the reflection indicates that what's a "growing" function on the original graph becomes a "widening" one on the inverse. Additionally, note that both functions are defined only for non-negative values, keeping them in the first quadrant.