[t] denotes the greatest integer function and [t−m]=[t]−m when m∈Z. If k=2[2x−1]−1 and 3[2x−2]+1=2[2x−1]−1 then the range of f(x)=[k+5x] is
∵[2x−2]‌=[(2x−1)−1] ‌=[2x−1]−1 ‌ let, ‌[2x−1]‌=a ⇒‌‌[2x−2]‌=a−1 ‌ so, ‌3(a−1)+1‌=2a−1 ⇒‌‌3a−3+1‌=2a−1⇒a=1 thus [2x−1]=1 ‌⇒‌‌1≤(2x−1)<2⇒2≤2x<3 ‌⇒‌‌1≤x<1.5 so, x∈[1,1.5) Now, k=2[2x−1]−1=2(1)−1=1 for x∈[1,1.5) f(x)‌=[k+5x]=[1+5x] ‌ at ‌‌x=1,f(x)=[6]=6‌‌(‌ minimum ‌) ‌ at ‌‌x=1.5,f(x)=[1+5×1.5] ‌=[8.5]=8‌‌‌ (maximum) ‌ so, f(x) can be 6,7,8 Hence, the range of f(x)={6,7,8}