Question

Suppose f is a continuous function defined for all real numbers which has a maximum value of 5 and a minimum value of -7. Which of the following must be true, which might be true, and which can never be true?
a) The maximum value of f(|x|) is 7.
b) The minimum value of f(|x|) is 0.
c) The maximum value of |f(x)| is 7.
d) The minimum value of |f(x)| is 5.

Answer 1

are both a and c supposed to be identical in the question?

Answer 2

no. a is the absolute value of x while c is the absolute value of y

Answer 3

So @CalcSOS what's your guess and reasoning for each? I can pretty quickly tell you why and what for each of them, but there's no sense in me just telling you. I'll help you understand. =D

Answer 4

I think (a) is never (b) is might (c) is must and (d) is never. I came to these conclusions by sketching possible graphs, but I don't understand the reasoning behind them

Answer 5

I see, well for the first one to be never you know that f(x) can be any number between -7 and 5. But since |x| is basically just choosing only x>0, it doesn't change the range at all.

Similarly for (c) you know that it can only ever take you to the max or min, never anything higher or lower than that. So if you have -7 as the lowest, you know that |-7|>5 so you can definitely say that it must.

For (b) we don't know because we just changed the domain from all the numbers to just the numbers greater than 0. We don't know where or when the max and min occur for what values of x, so it might be that 0 is the new minimum since -7 might happen when x<0. We'd need more information to really know this, like what the function is.

For (d) you know it has to be wrong because |f(x)| can easily be anything between 0 and 7, right? You could have had f(x)=3 since that's between -7 and 5 and then the absolute value of f(x) would just be 3, which is lower than 5, so 5 is never going to be the min here.

I hope that makes sense, since you got to the right answers you probably already know this on some level and hopefully this puts it into a little bit more common sense terms as to why. If you need me to clarify anything just ask and I'll try to rephrase it. =D

Answer 6

Your phrasing is spot on. This does clear things up, so I thank you for that! I guess my brain is just tired this late at night as i try to finish some homework problems. That is probably why I am not fully understanding the reasoning behind my actions lol