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7. Which if the following is always true of odd functions? I. f (−x) = −f(x) II. f(|x|) is even III. |f(x)| is even A. All of these are true. B. None of these are true. C. I only. D. II only. E. I and III only.

Mathematics
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What do you think?
I think all of them are true but not too sure. I get confused a lot with these odd and even function questions.
I will help you out. The definition of an odd function only tells us that a function \(f(x)\) is odd if it satisfies the property: \(f(-x)=-f(x)\), for all x in the domain of \(f\). Now we know that "I" is true, and we should use to make a conclusion about the other two statements. Would you like to try?

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Well wouldn't it still be positive because of the absolute values? And if it is not negative then it is not an odd function.
In II, we will only have \(f(|-x|)=f(|x|)=-f(|x|)\), and we can't say it's even. However in III, we have \(|f(-x)|=|-f(x)|=|f(x)|\), and therefore it's even.
Wait why is that? Because once we put the negative within the absolute values in "II", it should turn positive right?
Yes; \(|-x|=|x|\).
Then how is the outcome still negative?
It can be. Suppose you have \(f(x)=-x^3\). Then \(f(|-x|)=-(|-x|)^3=-x^3\). Right?
Oh so it doesn't matter if the absolute value only covers x? It can still be negative. But if the absolute value covers the whole function, then it will be positive? Correct?
Yes.

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