Which of the following functions has inverse function? a)`f: ZrarrZ`defined by`f(x)=x+2` b)`f: ZrarrZ`defined by`f(x)=2x` c)`f: ZrarrZ`defined byy`f(x)=x` d)`f: ZrarrZ`defined by`f(x)=|x|`

Explanation

`f:Z to Z ,f(x) = x+2` is both one-one and onto.
Hence invertible.
`f:Z to Z ,f(x) = 2x` is one-one but not onto as range is set
`{…,-4,-2,0,2,4,6, …}`. Hence not invertible.
`f:Z to Z ,f(x) = x` is one-one and onto, Hence invertible.
`f:Z to Z ,f(x) = |x|` is many-one, Also, range is
`{0,1,2,3,...}`, So, into, Hence not invertible.