Question

How do you determine if `f(x)=x/(x+1)` is an even or odd function?

Answer 1

`f(x)` is neither (not even or odd)

Explanation:

To determine if the given function is even or odd, you must prove the following cases:

`1`. If the function is even, then `f(-x)=f(x)`.
`2`. If the function is odd, then `f(-x)=-f(x)`.

Note that if the function is neither even nor odd, then it is neither.

Case 1 - Even Test
Determine the left and right sides of the given function using `f(-x)=f(x)`.

`"LS"=f(-x)color(white)(XXXXXXXXX)"RS"=f(x)`

`"LS"=((-x))/((-x)+1)color(white)(XXXXXXX)"RS"=color(blue)(|bar(ul(color(white)(a/a)color(black)(x/(x+1))color(white)(a/a)|)))`

`"LS"=color(blue)(|bar(ul(color(white)(a/a)color(black)((-x)/(-x+1))color(white)(a/a)|)))`

Since `"LS"` `!=` `"RS"`, `f(x)` is not even.

Case 2 - Odd Test
Determine the left and right sides of the given function using `f(-x)=-f(x)`.

`"LS"=f(-x)color(white)(XXXXXXXXX)"RS"=-f(x)`

`"LS"=((-x))/((-x)+1)color(white)(XXXXXXX)"RS"=-(x/(x+1))`

`"LS"=color(blue)(|bar(ul(color(white)(a/a)color(black)((-x)/(-x+1))color(white)(a/a)|)))color(white)(XXXXxx)"RS"=color(blue)(|bar(ul(color(white)(a/a)color(black)((-x)/(x+1))color(white)(a/a)|)))`

Since `"LS"` `!=` `"RS"`, `f(x)` is not odd.

The Conclusion
From the even and odd tests, since the given function is neither even nor odd, it is neither.

`:.`, `f(x)` is neither.