This is a fun problem because we are presented with a modified version of the cos(x) function, but as we will see, it is, in fact, not in any way different from the standard version.
cos|x| is the cos(x) function with the absolute value of x input into it. What this means is that if x >= 0 then it is replaced with x. If x < 0 then it is replaced with -x.
This means that our function is actually a two part piecewise function:
f(x) = {(cos|x| = cos(x) if x >= 0), (cos|x| = cos(-x) if x < 0) :}
However, cos is also an even function. For an even function, f(-x) = f(x).
This means that:
f(x) = {(cos|x| = cos(x) = cos(x) if x >= 0),
(cos|x| = cos(-x) = cos(x) if x < 0) :}
So cos|x| = cos(x).
The domain and range will be precisely the same as for the original function, that is: x can be all real numbers, and y will range from -1 to 1.