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#.
