Example 1 Given: \( f(x)=3 x^{2} \) 1.1 Determine the equation of the inverse of \( f(x) \) in the form \( y=\cdots \) 1.2 For what values of \( x \) is \( f(x) \) equal to its inverse? 1.3 Draw the graphs of \( f(x) \) and its inverse on the same set of axes. 1.4 Explain why the inverse is not a function. 1.5 How can you restrict the domain of \( f(x) \) so that its inverse is a function?

To find the inverse of the function \( f(x)=3x^{2} \), you start by replacing \( f(x) \) with \( y \): \[ y = 3x^2 \] Now, swap \( x \) and \( y \): \[ x = 3y^2 \] Solving for \( y \), we get: \[ y = \sqrt{\frac{x}{3}} \] So the inverse is: \[ f^{-1}(x) = \sqrt{\frac{x}{3}} \] For \( f(x) \) to equal its inverse, set \( 3x^{2} = \sqrt{\frac{x}{3}} \). Squaring both sides and simplifying gives solutions at \( x = 0 \) and \( x = \frac{1}{3} \) after checking the validity of each solution in both functions, confirming that this equation holds true. When graphing both functions, you will see \( f(x) \) is a parabola opening upwards, while \( f^{-1}(x) \) is a rightward-opening sideways parabola. They intersect at points like the origin and the value explained above, emphasizing symmetry. The inverse function isn’t a function itself since it fails the horizontal line test; every \( y \) value corresponds to more than one \( x \) value (negative and positive roots for \( y \)). Restricting the domain of \( f(x) \) to \( x \geq 0 \) or \( x \leq 0 \) allows the inverse to become a function since each \( y \) value will only correspond to one \( x \) value, making it valid. Making \( f(x) = 3x^2 \) a one-to-one function helps solve the inverse issue while maintaining the relationships intact.