The graph of a cubic function, y = f(x), is given below - HSC - SSCE Mathematics Extension 1 - Question 9 - 2023 - Paper 1
Question 9
The graph of a cubic function, y = f(x), is given below.
Which of the following functions has an inverse relation whose graph has more than 3 points with an x-coord... show full transcript
Worked Solution & Example Answer:The graph of a cubic function, y = f(x), is given below - HSC - SSCE Mathematics Extension 1 - Question 9 - 2023 - Paper 1
Step 1
A. y = f(√x)
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Answer
This function applies the square root to the input of the original function, which restricts the input and does not create more than three points with an x-coordinate of 1. Therefore, this option does not meet the criteria.
Step 2
B. y = 1/f(x)
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Answer
The reciprocal of the cubic function f(x) could potentially have a vertical asymptote depending on the roots of f. However, it does not guarantee more than three points with an x-coordinate of 1, as it would depend on the behavior of f(x) near that coordinate.
Step 3
C. y = f(|x|)
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Answer
This function reflects values for negative x coordinates onto the positive side. While it may introduce some symmetry, it does not strictly guarantee more than three points with an x-coordinate of 1.
Step 4
D. y = |f(x)|
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Answer
Taking the absolute value of the cubic function allows for the possibility of multiple points on the graph that could correspond to the same y-value, thus having more than three points that share the same x-coordinate of 1. This makes option D the correct choice.
