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 \( f(x) = \sqrt{x} \), we need to follow these steps: 1. **Find the Inverse Function**: The first step is to find the inverse of the function \( f(x) \). To do this, we replace \( f(x) \) with \( y \): \[ y = \sqrt{x} \] Next, we solve for \( x \) in terms of \( y \): \[ y^2 = x \quad \Rightarrow \quad x = y^2 \] Thus, the inverse function is: \[ f^{-1}(x) = x^2 \] 2. **Graph the Original Function**: The graph of \( f(x) = \sqrt{x} \) is a curve that starts at the origin (0,0) and increases to the right, approaching infinity as \( x \) increases. The domain of this function is \( x \geq 0 \) and the range is \( y \geq 0 \). 3. **Graph the Inverse Function**: The graph of the inverse function \( f^{-1}(x) = x^2 \) is a parabola that opens upwards. The domain of this function is \( x \geq 0 \) (since the output of the original function is non-negative), and the range is \( y \geq 0 \). 4. **Transformations**: The graph of the inverse function can be obtained from the graph of the original function by reflecting it across the line \( y = x \). This line acts as a mirror, and every point \( (a, b) \) on the original function corresponds to the point \( (b, a) \) on the inverse function. 5. **Summary of Transformations**: - The original function \( f(x) = \sqrt{x} \) is transformed into its inverse \( f^{-1}(x) = x^2 \) through reflection across the line \( y = x \). - The shape of the graph changes from a curve (the square root function) to a parabola (the square function). In conclusion, to graph the inverse of \( f(x) = \sqrt{x} \), we reflect the graph of the original function across the line \( y = x \), resulting in the graph of \( f^{-1}(x) = x^2 \).