f(x)=x raise to 2( x square)-3x+1,find k if f(2k)=2f(k) and f(2k)=f(k).

Dear Student,

f (x) = x² -3x +1

=> f (2k) = (2k)² - 3(2k) + 1

=> f (2k) = 4k² - 6k + 1

And f (k) = k² -3k +1

As, f (2k) = 2f (k)

=> 4k² - 6k + 1 = 2k² -6k +2

=> 2k² -1 = 0

=> k² = 1/2

=> k = 1/√2 or -1/√2

And for f (2k) = f (k)

i.e. 4k² - 6k + 1 = k² -3k +1

=> 3k² - 3k = 0

=> 3k (k-1) = 0

=> k = 0 or 1.

Regards Topperlearning.