Question

Salut am de la exercitul 33 pana la 37 va rog frumos sa ma ajutati cu ce puteti.Multumesc mult❤️

Answer 1

33.

$f,g : \mathbb{R} \to \mathbb{R}, f(x)=3x+5, g(x)= -x+4\\(f\circ g)(x) = f(g(x)) = f(-x+4) = 3(-x+4) + 5 = -3x + 12 + 5 = -3x + 17\\(g\circ f)(x) = g(f(x)) = g(3x+5) = -(3x+5) + 4 = -3x - 5 + 4 = -3x - 1\\(f\circ f)(x) = f(f(x)) = f(3x+5) = 3(3x+5) + 5 = 9x + 15 + 5 = 9x + 20\\(g\circ g)(x) = g(g(x)) = g(-x+4) = -(-x+4) + 4 = x \cancel{- 4} \cancel{+ 4} = x$

34.

$f,g : \mathbb{R} \to \mathbb{R}, f(x)=2x-3, g(x)= x^2-4x = x(x-4)\\(f\circ g)(x) = f(g(x)) = f(x(x-4)) = 2\cdot x(x-4) - 3 = 2x^2 - 8x - 3\\(g\circ f)(x) = g(f(x)) = g(2x-3) = (2x-3)(2x-3-4) = (2x-3)(2x-7) = 4x^2 -14x - 6x + 21 = 4x^2 -20x + 21\\(f\circ f)(x) = f(f(x)) = f(2x-3) = 2(2x-3) - 3 = 4x - 6 - 3 = 4x - 9\\(g\circ g)(x) = g(g(x)) = g(x(x-4)) = (x(x-4))(x(x-4) - 4) = (x^2 - 4x)(x^2 - 4x - 4) = x^4 - 4x^3 - 4x^2 - 4x^3 + 16x^2 + 16x = x^4 - 8x^3 + 12x^2 + 16x$

35.

$f,g : \mathbb{R} \to \mathbb{R}, f(x)=4x-5, g(x)= 3x-8\\(f\circ g)(x) = 11\\f(g(x)) = 11\\ f(3x-8) = 11\\ 4(3x-8) - 5 = 11\\ 4(3x-8) = 16\\ 3x-8 = 4\\ 3x = 12\\ x = 4\\(g\circ f)(x) + 35 = 0\\ g(f(x)) = -35 \\ g(4x-5) = -35\\ 3(4x-5) - 8 = -35\\ 3(4x-5) = -27\\4x-5 = -9\\ 4x = -4\\ x = -1$

36.

$f : \mathbb{R} \to \mathbb{R}, f(x)=mx+5, m \in \mathbb{R}\\(f\circ f)(-1) = 11\\f(f(-1)) = 11\\ f(m\cdot (-1) + 5) = 11\\ f(-m+5) = 11\\ m(-m+5)+5 = 11\\ -m^2 + 5m + 5 = 11\\ -m^2 + 5m - 6 = 0\\m^2 - 5m + 6 = 0\\(m-3)(m-2) = 0\Rightarrow m \in \{2,3\}$

37.

$f : \mathbb{R} \to \mathbb{R}, f(x)=x+2\\ f(f(x)) = f^2(x)\\ f(x+2) = (x+2)^2\\x+2+2 = (x+2)^2\\ x\cancel{+4} = x^2 + 4x \cancel{+ 4}\\x = x^2 + 4x\\ x^2 + 3x = 0\\x(x+3) = 0\implies x \in \{-3, 0\}$