(d) Given the function \( f(x)=-\sqrt{3 x} \). (1) Write down the domain and range of \( f \). (2) Determine the equation of \( f^{-1} \). (3) Sketch the graphs of \( f \) and \( f^{-1} \) on the same set of axes. (4) Write down the equation of (i) \( g \), the reflection of \( f \) in the line \( x=0 \). (ii) \( h \), the reflection of \( f \) in the line \( y=0 \).

The function \( f(x) = -\sqrt{3x} \) has a domain of \( x \geq 0 \) because the square root function is only defined for non-negative values. The range of \( f \) is \( y \leq 0 \) since it outputs non-positive values due to the negative sign. To find the inverse function, we start with \( y = -\sqrt{3x} \). Solving for \( x \), we square both sides to get \( y^2 = 3x \), or \( x = \frac{y^2}{3} \). Replacing \( y \) with \( x \), the inverse function is: \( f^{-1}(x) = \frac{x^2}{3} \). For graphing, the graph of \( f \) is a downward-opening curve starting at the origin and extending left. The inverse \( f^{-1} \) is a parabola opening upwards, starting at the origin as well. (i) The reflection of \( f \) in the line \( x=0 \) flips it to the other side of the y-axis, leading to \( g(x) = -\sqrt{-3x} \) for \( x \leq 0 \). (ii) Reflecting \( f \) in the line \( y=0 \) will change the sign of the function, resulting in \( h(x) = \sqrt{3x} \) for \( x \geq 0 \). Remember, reflections are like taking a funhouse mirror version of the original graph—always cool to see how it changes!