Find the sum ofdifference of the identity function and the modulus fun – InterviewSolution

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1.	Find the sum ofdifference of the identity function and the modulus function.
Answer» Let `f:RtoR:f(x)=x` be the identity function. And, let `g:RtoR:g(x)=\|x\|` be the modulus function. Then, dom (f) = R and dom (g)=R. `:."dom "(f)nn"dom "(g)=RnnR=R`. (i) dom `(f+g)="dom "(f)nn"dom "(g)=R`. Now, `(f+g):RtoR` is given by `(f+g)(x)=f(x)+g(x)` `=x+\|x\|=x+{(x",when "xge0),(-x",when "xlt0):}` `={{:(x+x",when "xge0),(x-x",when "lt0):}={{:(2x",when "xge0),(0",when "xlt0):}` Hence, `(f+g)(x)={{:(2x",when "xge0),(0",when "xlt0):}` (ii) dom `(f-g)="dom "(f)nn"dom "(g)=R`. `:.(f-g)(x)=f(x)=g(x)` `=x-\|x\|=x-{{:(x",when "xge0),(-x",when "xlt0):}` `={{:(x-x",when "xge0),(x+x",when "xlt0):}={{:(0",when "xge0),(2x",when "xlt0):}` `:.(f-g)(x)={{:(0",when "xge0),(2x",when "xlt0):}`

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