f-f-x-ax-b-1-show-that-f-ax-b-af-x-b-deduce-f-ax-b-2-Show-that-f-x-is-a-constant-hence-deduce-f-

Question Number 194282 by alcohol last updated on 02/Jul/23

f (f (x)) = a x + b

1 . s h o w t h a t f (a x + b) = a f (x) + b

d e d u c e f^{'} (a x + b)

2 . S h o w t h a t f^{'} (x) i s a c o n s t a n t

h e n c e d e d u c e f

Answered by Frix last updated on 02/Jul/23

f (x) = α x + β \Rightarrow

f (f (x)) = α^{2} x + (α + 1) β \Rightarrow

\underset{[\Rightarrow a > 0]}{a = α^{2}} \land b = (α + 1) β \Leftrightarrow α = \pm \sqrt{a} \land β = \frac{b}{1 \pm \sqrt{a}} \Rightarrow

f (x) = \pm \sqrt{a} x + \frac{b}{1 \pm \sqrt{a}}

f (a x + b) = f (α^{2} x + (α + 1) β) =

= α^{3} x + (α^{2} + α + 1) β

a f (x) + b = α^{2} (α x + β) + (α + 1) β =

= α^{3} x + (α^{2} + α + 1) β

f^{'} (a x + b) = α^{3} = \pm a^{\frac{3}{2}}

f^{'} (x) = \pm \sqrt{a}

Facebook Tweet Pin