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##Pyth, 9 8 bytes

Thanks to @isaacg for -1 byte!

sXz.HM16

Subtracting a certain color's value from 255 is equivalent to subtracting each of its hexadecimal digits from 15. Say a number is 16a+b. Then the value of the number created by subtracting its digits from 15 is 16(15-a) + (15-b) = 255 - (16a+b).

sXz.HM16     implicit: z=input()
      16      
   .HM        map hex representation over range
   .HM16     '0123456789abcdef'
  z           the input string
 X            Translate characters in x1 present in x2 to reversed x2
              that is, '0' becomes f, '1' becomes 'e', and so on.
              The initial '#' is unchanged.
s             That produced a list, so join into a string by reducing +

Try it here. Test suite.

Pyth, 9 8 bytes

Thanks to @isaacg for -1 byte!

sXz.HM16

Subtracting a certain color's value from 255 is equivalent to subtracting each of its hexadecimal digits from 15. Say a number is 16a+b. Then the value of the number created by subtracting its digits from 15 is 16(15-a) + (15-b) = 255 - (16a+b).

sXz.HM16     implicit: z=input()
      16      
   .HM        map hex representation over range
   .HM16     '0123456789abcdef'
  z           the input string
 X            Translate characters in x1 present in x2 to reversed x2
              that is, '0' becomes f, '1' becomes 'e', and so on.
              The initial '#' is unchanged.
s             That produced a list, so join into a string by reducing +

Try it here. Test suite.

##Pyth, 9 bytes

sXz.HMU16

Subtracting a certain color's value from 255 is equivalent to subtracting each of its hexadecimal digits from 15. Say a number is 16a+b. Then the value of the number created by subtracting its digits from 15 is 16(15-a) + (15-b) = 255 - (16a+b).

sXz.HMU16     implicit: z=input()
      U16     range(16)
   .HM        map hex representation
   .HMU16     '0123456789abcdef'
  z           the input string
 X            Translate characters in x1 present in x2 to reversed x2
              that is, '0' becomes f, '1' becomes 'e', and so on.
              The initial '#' is unchanged.
s             That produced a list, so join into a string by reducing +

Try it here. Test suite.

##Pyth, 9 8 bytes

Thanks to @isaacg for -1 byte!

sXz.HM16

Subtracting a certain color's value from 255 is equivalent to subtracting each of its hexadecimal digits from 15. Say a number is 16a+b. Then the value of the number created by subtracting its digits from 15 is 16(15-a) + (15-b) = 255 - (16a+b).

sXz.HM16     implicit: z=input()
      16      
   .HM        map hex representation over range
   .HM16     '0123456789abcdef'
  z           the input string
 X            Translate characters in x1 present in x2 to reversed x2
              that is, '0' becomes f, '1' becomes 'e', and so on.
              The initial '#' is unchanged.
s             That produced a list, so join into a string by reducing +

Try it here. Test suite.

##Pyth, 9 bytes

sXz.HMU16

Subtracting a certain color's value from 255 is equivalent to subtracting each of its hexadecimal digits from 15. Say a number is 16a+b. Then the value of the number created by subtracting its digits from 15 is 16(15-a) + (15-b) = 255 - (16a+b).

sXz.HMU16     implicit: z=input()
      U16     range(16)
   .HM        map hex representation
   .HMU16     '0123456789abcdef'
  z           the input string
 X            Translate characters in x1 present in x2 to reversed x2
              that is, '0' becomes f, '1' becomes 'e', and so on
s             That produced a list, so join into a string by reducing +

Try it here. Test suite.

##Pyth, 9 bytes

sXz.HMU16

Subtracting a certain color's value from 255 is equivalent to subtracting each of its hexadecimal digits from 15. Say a number is 16a+b. Then the value of the number created by subtracting its digits from 15 is 16(15-a) + (15-b) = 255 - (16a+b).

sXz.HMU16     implicit: z=input()
      U16     range(16)
   .HM        map hex representation
   .HMU16     '0123456789abcdef'
  z           the input string
 X            Translate characters in x1 present in x2 to reversed x2
              that is, '0' becomes f, '1' becomes 'e', and so on.
              The initial '#' is unchanged.
s             That produced a list, so join into a string by reducing +

Try it here. Test suite.