Assign a grade of A (correct) , C (partially correct) , or F (failure) to each. Justify assignments of grades other than A. (a) Claim. The functions f and g are equal, Where f and g are given by f(x) = ∣x∣x​ and g(x) ={ 1−1​ if x≥0 if x<0​ "Proof. " Let x be a real number. If x is positive, then ∣x∣x​ = xx​ =1, so f(x) =g(x) . If x is negative, then ∣x∣x​ = −xx​ =−1, so f(x) =g(x) , In every case,f(x) =g(x) , so f=g. (b) Claim. The functions f(x) =1+ x1​ and g(x) = xx+1​ are equal. "Proof. " The domain of each function is assumed to be the largest possible subset of R. Thus, Dom(f) =Dom(g) =R−{0}. For every x∈R−(0) , we have f(x) =1+ x1​ = xx​ + x1​= xx+1​=g(x)