The graph of a cubic function, y = f(x), is given below - HSC - SSCE Mathematics Extension 1 - Question 9 - 2023 - Paper 1

To determine which function has an inverse relation with more than three x-coordinates at x = 1, we need to analyze each option:

1. Option A: y = f(x) - This is the original cubic function. For a cubic function, the graph will only intersect the vertical line (x=1) at most three times. Therefore, this option does not meet the requirement.

2. Option B: y = 1/f(x) - Since the function is in the form of 1 divided by a cubic function, this option may introduce vertical asymptotes. However, it won't guarantee more than three intersections at x=1 and may even result in undefined points, so it does not fulfill the requirement either.

3. Option C: y = f(|x|) - This transformation reflects the graph of f(x) for x < 0 to the positive side, creating symmetry. While this can increase the number of points at any given x-coordinate, it still tends toward having a maximum of three unique intersections.

4. Option D: y = |f(x)| - This transformation takes the absolute value of the cubic function, thus making all y-values non-negative. Consequently, depending on the cubic function's maximum and minimum points, this could result in multiple intersections at x=1, potentially exceeding three points.

After evaluating each option, the correct answer is D. y = |f(x)|, as it can yield more than three graph points with an x-coordinate of 1.