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y-x-2-x-R-y-Range-




Question Number 193971 by mnjuly1970 last updated on 24/Jun/23
        y= ⌊ x^( 2)  ⌋ + ⌊ x ⌋               ⇒  R_y  =?       Range
y=x2+xRy=?Range
Answered by MM42 last updated on 24/Jun/23
R_y ={−1,0,1,2,3,...}
Ry={1,0,1,2,3,}
Answered by mahdipoor last updated on 24/Jun/23
get    x=(√(n^2 +m))      n∈W,0≤m∈R<2n+1  [x^2 ]+[x]=n^2 +[m]+n         0≤[m]<2n+1  R_(y1) ={n^2 +n+m∣n,m∈W,m≤2n}  ,  get    x=−(√(n^2 +m))≠0      n∈W,0≤m∈R<2n+1  [x^2 ]+[x]=n^2 +[m]−n−1        0≤[m]<2n+1  R_(y2) ={n^2 −n−1+m∣n,m∈W,m≤2n}  ,  ∩⇒R_y ={r∈Z∣n^2 −n−1≤r≤n^2 +3n,n∈W}  for ∀n∈W   n^2 +3n≥(n+1)^2 −(n+1)−1  ⇒⇒R_y ={−1,0,1,2,...,∞}
getx=n2+mnW,0mR<2n+1[x2]+[x]=n2+[m]+n0[m]<2n+1Ry1={n2+n+mn,mW,m2n},getx=n2+m0nW,0mR<2n+1[x2]+[x]=n2+[m]n10[m]<2n+1Ry2={n2n1+mn,mW,m2n},Ry={rZn2n1rn2+3n,nW}fornWn2+3n(n+1)2(n+1)1⇒⇒Ry={1,0,1,2,,}

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