Let f(x) be any function with domain all real numbers - HSC - SSCE Mathematics Advanced - Question 9 - 2023 - Paper 1
Question 9
Let f(x) be any function with domain all real numbers.
Which of the following is an even function, regardless of the choice of f(x)?
A. 2f(x)
B. f(f(x))
C. (f(-x))... show full transcript
Worked Solution & Example Answer:Let f(x) be any function with domain all real numbers - HSC - SSCE Mathematics Advanced - Question 9 - 2023 - Paper 1
Step 1
A. 2f(x)
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Answer
This function is not necessarily even because multiplying by 2 does not ensure that f(x) = f(-x).
Step 2
B. f(f(x))
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Answer
The composition of a function does not guarantee that it will retain the property of being even, as f(f(-x)) may not equal f(f(x)).
Step 3
C. (f(-x))^2
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Answer
This expression produces a non-negative value, yet it does not automatically make it even unless f(-x) has specific properties.
Step 4
D. f(x)f(-x)
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Answer
This function is even because it satisfies the definition of an even function: f(x)f(-x) = f(-x)f(x). Therefore, the product of any function and its reflection across the y-axis will always yield an even function.
