(a) Given \( f(x)=2 x^{2} \). (1) Restrict the domain of \( f \) in two different ways so that the inverse is a (2) Hence sketch the graph of \( f \) and its inverse for each restriction of the (3) Write the equation of the inverse function in the form \( f^{-1}(x)=\ldots \)

To ensure that the inverse function is indeed a function, we can restrict the domain of \( f(x) = 2x^2 \) in two different ways: 1. **Restriction 1**: Limit the domain to \( x \geq 0 \). This will consider the right side of the parabola, where the function is increasing. 2. **Restriction 2**: Limit the domain to \( x \leq 0 \). This will focus on the left side of the parabola, where again the function is increasing. Next, let’s sketch the graphs. For **Restriction 1** \( (x \geq 0) \): - The graph of \( f(x) = 2x^2 \) is a right-opening parabola starting at the origin and moving upward. The inverse graph will be a sideways parabola that opens to the right, which reflects the positive outputs of \( f \). For **Restriction 2** \( (x \leq 0) \): - The graph of \( f(x) = 2x^2 \) reflects the same parabola but we only consider the left half. The inverse graph will be a sideways parabola opening to the left. Now, let’s find the equations of the inverse functions. For **Restriction 1**: 1. Set \( y = 2x^2 \) and solve for \( x \): \[ x = \sqrt{\frac{y}{2}} \] 2. Thus, the inverse function is: \[ f^{-1}(x) = \sqrt{\frac{x}{2}} \] For **Restriction 2**: 1. Similarly, setting \( y = 2x^2 \) and solving gives: \[ x = -\sqrt{\frac{y}{2}} \] 2. Hence, the inverse function is: \[ f^{-1}(x) = -\sqrt{\frac{x}{2}} \] And there you have it! Your inverses are ready to rock and roll!