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 12 over x

Answer 1

y = x + 4/x

replace x with -x. Do you get back the original equation after simplifying. if you do, the function is even. replace y with -y AND x with -x. Do you get back the original equation after simplifying. If you do, the function is odd. A function can be either even or odd but not both. Or it can be neither one. Let's first replace x with -x

y = -x + 4/-x = -x - 4/x = -(x + 4/x)

we see that this function is not the same because the original function has been multiplied by -1 . Let's replace y with -y and x with -x

-y = -x + 4/-x

-y = -x - 4/x

-y = -(x + 4/x)

y = x + 4/x

This is the original equation so the function is odd.

Answer 2

Neither even nor odd.

Explanation:

A functio f(x) is even if f(-x)=f(x) for all x in the domain.

A function f(x) is odd if f(-x)=-f(x) for all x in the domain.

First note that

`f(x)=(x+12)/(x)`

hence

`f(-x)=(-x+12)/(-x)=((-1)(x-12))/((-1)x)=((x-12))/(x)\\eq f(x)`

which tells us that the function is not even.

On the other hand,

`-f(x)=-(x+12)/(x)=(-x-12)/(x)\\eq f(-x)`

which tells us that the function is not odd.

Therefore, f(x) is neither even nor odd.