Determine which of the following are in the same Thetaclass

Determine which of the following are in the same Theta-class. A function may be in a class by itself. f_1(n) = 5 nlg(n), f_2(n) = 6n^2 - 3n + 7, f_3(n) = 1.5^n, f_4(n) = lg (n^4), f_5(n) = 13, 463, f_6(n) = -15 n, f_7 (n) = lg (lg(n)), f_8 (n) = 9n^0.7, f_9(n) = n!, f_10(n) = n + lg (n), f_11(n) = Squareroot n + 12n, f_12 (n) = lg (n!)

Solution

f₁(n) = 5nlg(n) = ( nlg(n) )

f₂(n) = 6n²-3n+7 = ( n² )

f₃(n) = 1.5ⁿ = ( 1.5ⁿ )

f₄(n) = lg(n⁴) = ( lg(n) )

f₅(n) = 13463 = ( 1 )

f₆(n) = -15n = ( n )

f₇(n) = lg(lg(n)) = ( lg(lg(n)) )

f₈(n) = 9n^{0.7 =} ( n^0.7 )

f₉(n) = n! = ( n! )

f₁₀(n) = n+lg(n) = ( n )

f₁₁(n) = n^0.5 + 12n = ( n )

f₁₂(n) = lg(n!) = ( nlg(n) )

( n ) : f₆ , f₁₀ , f₁₁

( nlg(n) ) : f₁ , f₁₂

( lg(n) ) : f₄

( 1 ) : f₅

( lg(lg(n)) ) : f₇

( 1.5ⁿ ) : f₃

( n² ) : f₂

( n^0.7 ) : f₈

( n! ) : f₉