Question

dadas as funções g(x) = 2 x+1 e h (x) = x+2

g(h(x)) = g(g)(x))

Answer 1

 g(g(x)) = g(2x + 1) = 2*(2x + 1) + 1 = 4x + 2 + 1 = 4x + 3 

g(f(x)) = g(x² - 1) = 2*(x² - 1) + 1 = 2x² - 2 + 1 = 2x² - 1 

h(g(x)) = h(2x + 1) = (2x + 1 - 1)/(2(2x + 1) + 3) = 2x/(4x + 2 + 3) = 2x/(4x + 5) 

f(g(x)) = f(2x + 1) = (2x + 1)² - 1 = 4x² + 4x + 1 - 1 = 4x² + 4x 

h(h(x)) = h((x - 1)/(2x + 3)) = ((x - 1)/(2x + 3) - 1)/(2(x - 1)/(2x + 3) + 3) 
= ((x - 1 - 2x - 3)/(2x + 3))/((2x - 2 + 6x + 9)/(2x + 3)) 
= ((-x - 4)/(2x + 3))/((8x + 7)/(2x + 3)) 
=(-x - 4)/(8x + 7) 

f(g(m)) = 4m² + 4m = 0 
m(4m + 1) = 0 
m' = 0 
m'' = -1/4