SOLUTION: Do functions have to be even or odd, necessarily? i have this question here for a take home quiz, and thanks everyone in advance for all of your help. i think its wonderful you vo

Question 82645: Do functions have to be even or odd, necessarily? i have this question here for a take home quiz, and thanks everyone in advance for all of your help. i think its wonderful you volunteer your time. maybe some of the students will grow up to study math because they'll find out how fun it really is! :-)
1) the function f(x)= x - |x| is:
a) an odd function
b) an even function
c) neither
when i work out the problem i can't make it look even or odd, so is it neither? i thought the function HAS to be either even or odd? can someone tell me the answer and clarify this, or let me know if i got it right? thanks!
Also, another question i answered, but not confident i got correct is: the domain for the funcion [f/g](x) with f(x)= x-3 and g(x) = (square root of (x+4)) is:
a) is [-4, infinity)
b) is [4, infinity)
c) none of the above
is this correct: g(x) is equal to all real numbers except for (-4) and f(x) is equal to all real numbers except for (3). so i guess this is all of the above, but how would you right this answer in the correct notation? and if i'm wrong, please explain what the right answer is. thank you again for your help, its really appreciated, i'm making a donation to your site when i get home from work tonight, i'm really impressed with the quality of the tutors here, i didn't think someone who was really good would answer the questions for free, but i was wrong. math people are great people!!! :-)
Answer by kev82(151) (Show Source): You can put this solution on YOUR website!
First question: You are correct. The function is neither even or odd. there are many functions like this. For example f(x)=x-1.

There are many ways to construct such functions. Any odd function plus a (non-zero) constant will be neither odd or even. Can you explain why?

In fact an even function plus and odd function will be neither even or odd (apart from one special case) can you think what this case is?

Second question: I don't understand the notation you're using. I haven't done this kind of maths in years. Do you mean:

Well, the answer depends on what sets we're working with, but I'm gonna guess that you're working with real numbers.

The way I would tackle a question like this, is to go through these steps:

1) Identify which operations you are doing which can make things go bad.

2) For each operation that can go bad, work out at which points it goes bad.

3) Take these bad points out of

(The real numbers

), and that leaves you the domain.

Well, the (possibly) bad operations you have here are the division and the square root.

The division will go bad if we divide by zero, so that's when x=-4. The square root will go bad if the inside is negative, and that's when

.

So the bad points are when

. Taking this away from

leaves the domain as:

Note: this isn't one of the choices you've been given. It is open on the left, whereas the choice was closed on the left.

Hope that helps,
Kev
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