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
The function y = f(x) is a cubic function, which is continuous and polynomial. It has a one-to-one relationship only at certain intervals and does not generally have more than 3 points corresponding to the same x-coordinate.
Step 2
B. y = 1/f(x)
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Answer
The function y = 1/f(x) is the reciprocal of the cubic function. This can introduce undefined points where f(x) = 0, which will limit the number of occurrences of x-coordinates and does not guarantee more than 3 points at x = 1.
Step 3
C. y = f(|x|)
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Answer
The function y = f(|x|) reflects the portion of the cubic function on the positive side to the negative side. While creating a symmetrical property, it typically keeps the number of solutions for any specific value of x less than or equal to the original cubic function.
Step 4
D. y = |f(x)|
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Answer
The function y = |f(x)| takes the absolute value of the output of the cubic function. This alteration ensures that negative outputs are reflected above the x-axis, potentially resulting in more than 3 points with an x-coordinate of 1, as both positive and negative outputs will be represented.
