Quick theory overview:

For a function \(\mathcal{f}\) that maps outcomes \(\omega\) from a sample space \(\Omega\) to the real numbers, the inverse image of a set \(B\) is the collection of all outcomes that map into \(B\). We are interested in sets of the form \(B = (-\infty, c]\). Therefore, the inverse image is:

\(f^{-1}(-\infty, c]) = \{ \omega_i \in \Omega \mid f(\omega_i) \leq c\}\)

Procedure:

• Enter a value for the constant 'c' in the input box.
• Press the "Generate Function Values" button. This will create a new random function \(\mathcal{f}\) and display its output values.
• The plot will visualize the function values and show the threshold 'c' as a horizontal line.
• Based on the definition, determine which outcomes \(\{\omega_1, \omega_2, \omega_3, \omega_4\}\) have function values less than or equal to 'c'.
• Select the correct set from the grid of possible answers.
• The "Observations" panel will provide immediate feedback on your selection.