Let `f(x)=[9^x-3^x+1]` for all `x in (-oo, 1),` then the range of `f(x)` is, ([.] denotes the greatest integer function).

Options

A. `{0, 1, 2, 3, 4, 5, 6, 7}`
B. `{0, 1, 2 , 3, 4, 5, 6}`
C. `{1, 2, 3, 4, 5, 6, 7}`
D. `{1, 2, 3, 4, 5, 6}`

Answer

D

Explanation

We have ,
`f(x) = [9^(x0 - 3^(x) + 1]=[(3^(x)-(1)/(2))^(2) +(3)/(4)]`
For `x in (-oo,1]`, we observe that `g(x) = (3^(x)-(1)/(2))^(2)+(3)/(4)` is an increasing funtion of x. Therefore,
Therefore, range of `f(x) = [g(x)] = { 1,2,3,4,5,6}`
Hence, option (d) is correct.