Question

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

Answer 1

Relate `f(-x)` to `f(x)`

Explanation:

`f(-x) = -x + abs(-x)`

`= -x + abs(x)`

Since `f(-x) != f(x)`, `f(x)` is not an even function.
Since `f(-x) != -f(x)`, `f(x)` is not an odd function.

Here is a graph of `y = f(x)`.
graph{x+abs(x) [-10, 10, -5, 5]}
If `f(x)` is an even function, the `y`-axis would be a line of symmetry.

If `f(x)` is an odd function, the graph would have rotational symmetry about the origin.

These are graphical methods to check whether a function is odd or even. However neither of the symmetries are present.

Hence, `f(x)` is neither an odd function nor an even function.

Answer 2

Neither.

Explanation:

`f(-x) = -x + |x|` is neither f(x) nor -f(x).
`|x|=-x, x<0` and `|x|=x, x>0`.
`f(x)=x-x=0, x<=0`..
`f(x)=2x, x>0`..
The graph for y = f(x) comprises the negative x-axis continued as the straight line y = 2x in the first quadrant.,