Question

Find the sum and product of identity function and the modulus function.

Answer 1

Let `f:RtoR:f(x)=xandg:RtoR:g(x)=|x|` be the identity function and the modulus function respectively.
Then, dom `(fg)="dom "(f)nn"dom "(g)=RnnR=R`.
`(fg):RtoR:(fg)(x)=f(x).g(x)`.
Now, `(fg)(x)=f(x),g(x)=x.|x|`
`=x.{{:(x",when "xge0),(-x" ,when "lt0):}={{:(x^(2)",when "xge0),(-x^(2)",when "xlt0):}`
Hence, `(fg)(x)={{:(x^(2)",when "xge0),(-x^(2)",when "xlt0):}`.