Question

Va rog cât mai complet și corect ​

Answer 1

Răspuns:

Explicație pas cu pas:

f o g = f(g (x))

deci in f inlocuiesti pe x ci g(x)

f(x) = x^2 + 3x

g(x) = -x + 1

f o g = (-x + 1)^2 + 3*(-x + 1) = x^2 - 2x + 1 - 3x + 3 = x^2 - 5x + 4

g o f = g(f(x)) = -(x^2 + 3x) + 1 = -x^2 - 3x + 1

f o f = f(f(x)) = (x^2 + 3x)^2 + 3*(x^2 + 3x) = x^4 + 6x^3 + 9x^2 + 3x^2 + 9x

= x^4 + 6x^3  + 12x^2 + 9x

g o g = g(g(x)) = - (-x + 1) + 1 = x - 1 + 1 = x

Answer 2

$\displaystyle f,g:\mathbb{R}\rightarrow\mathbb{R},~f(x)=x^2+3x,~g(x)=-x+1\\\\ f\circ g,~g\circ f,~f\circ f, g\circ g=?\\\\ (f\circ g)(x)=f(g(x))=(g(x))^2+3g(x)=(-x+1)^2+3(-x+1)=\\\\=(-x)^2+2 \cdot (-x) \cdot 1+1^2-3x+3=x^2-2x+1-3x+3=x^2-5x+4$

$(g\circ f)(x)=g(f(x))=-f(x)+1=-\Big(x^2+3x\Big)+1=-x^2-3x+1$

$(f\circ f)(x)=f(f(x))=(f(x))^2+3f(x)=\Big(x^2+3x\Big)^2+3\Big(x^2+3x\Big)=\\\\=\Big(x^2\Big)^2+2 \cdot x^2 \cdot 3x+(3x)^2+3x^2+9x=x^4+6x^3+9x^2+3x^2+9x=\\\\=x^4+6x^3+12x^2+9x$

$(g\circ g)(x)=g(g(x))=-g(x)+1=-(-x+1)+1=x-1+1=x$