Odd, Even, or Neither: Combining Functions in Algebra

[torquerotates]

Homework Statement

Consider two functions, either of which is even or odd. But neither is neither even nor odd. Determine which algebraic combinations(sum, difference, quotient, product) of the given functions will result in an odd function, an even function, and in a function that is neither even nor odd.

Homework Equations

f(-x)=f(x) for even and f(-x)=g(x) for odd.

The Attempt at a Solution

I don't know how to unravel the logic. Say I pick the first function, it could be even or odd. But it is not (not even and not odd)=even or odd. So it must be even or odd? What is the point in all this? Are we not back at the starting point?

 

[Student100]

I think you're getting caught up in the wording, it's basically asking you to pick combinations of even and odd functions and describe their output composed with various operators.

Even - Even
Odd - Odd
Odd - Even
Even -Odd

 

[Mark44]

Homework Statement

Consider two functions, either of which is even or odd. But neither is neither even nor odd. Determine which algebraic combinations(sum, difference, quotient, product) of the given functions will result in an odd function, an even function, and in a function that is neither even nor odd.

The wording in the second sentence is unwieldy. The two functions are either even or odd. You don't have to consider any functions that don't fall into one of these categories.

Homework Equations

f(-x)=f(x) for even and f(-x)=g(x) for odd.

Your definition for an odd function is incorrect. What's the correct definition?

The Attempt at a Solution

I don't know how to unravel the logic. Say I pick the first function, it could be even or odd. But it is not (not even and not odd)=even or odd. So it must be even or odd? What is the point in all this? Are we not back at the starting point?

Consider one arithmetic operation at a time. Call the functions e and o. As Student100 says, look at the four possible combinations and say whether the sum is even, odd, or neither.

Do the same thing for each of the remaining three arithmetic operations.

 

[torquerotates]

Thanks, I was tired, f(-x)=-f(x) for odd.

product(-x)=e(-x)o(-x)=(e(x))(-o(x))=-e(x)o(x)=-product(x) =>product(x) is odd

quotient(-x)=e(-x)/o(-x)=e(x)/(-o(x))=-e(x)/o(x)=-quotient(x) => quotient(x) is odd.

The sum and difference gives us neither.

There is nothing that can give an even.