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.... 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
An even function satisfies the condition f(−x)=f(x). Here, 2f(x) does not necessarily satisfy this condition because if f(x) is odd, then 2f(−x)=−2f(x), which violates the requirement.
Step 2
B. $f(f(x))$
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Answer
The function f(f(x)) does not generally satisfy the even function property. If f(x) is odd, then f(f(−x)) may not equal f(f(x)).
Step 3
C. $(f(-x))^2$
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Answer
The expression (f(−x))2 is always non-negative and satisfies the condition of being even since (−x)2=x2. Hence, it is an even function.
Step 4
D. $f(x)f(-x)$
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Answer
The product f(x)f(−x) is also always even. This is because:
f(−x)f(−(−x))=f(−x)f(x)
Thus, this expression satisfies the condition for being an even function.
