Question

Determine algebraically whether the function is even, odd, or neither even nor odd. f as a function of x is equal to x plus quantity 4 over x.

Answer 1

1) Even for first problem.

2) Neither for the second.

I really think the problem I labeled 1 is the correct interpretation but just in case you meant the latter I wrote the latter as well. The sentence translates to exactly what I have for problem number 1.

The Problem:

1) Determine if
`f(x)=x+(4)/(x)` is even, odd, or neither.

2) Determine if
`f(x)=(x+4)/(x)` is even, odd, or neither.

Explanation:

`f(-x)=f(x)` implies
`f` is even.

`f(-x)=-f(x)` implies
`f` is odd.

So either definition says we have to plug in
`-x`.

1)

`f(x)=x+(4)/(x)` with new input
`-x`:

`f(-x)=-x+(4)/(-x)`

`f(-x)=-x+-(4)/(x)`

`f(-x)=-(x+(4)/(x))`

`f(-x)=-f(x)`

This means
`f` is even since we got the same thing we started with.

2)

`f(x)=(x+4)/(x)` with new input
`-x`:

`f(-x)=(-x+4)/(-x)`

`f(-x)=(-(x-4))/(-x)`

`f(-x)=(x-4)/(x)`

This is neither the same or the opposite of what we started with.