Question

Ex 3(a) și ex 5(a) va rog dau coroană ​

Answer 1

Răspuns:

Explicație pas cu pas:

3) f,g : R --> R

a) f(x) = 2x-1 ; g(x) = 3x-2

(f o f)(x) = f(f(x)) = 2(2x-1)-1 = 4x-2-1 = 4x-3

f(x) = (f o f)(x) => 2x-1 = 4x-3 => 2x = 2 => x = 1

---

(g o f)(x) = g(f(x)) = 3(2x-1)-2 = 6x-3-2 = 6x-5

(f o g)(x) = f(g(x)) = 2(3x-2)-1 = 6x-4-1 = 6x-5

(g o f)(x) = 2(f o g)(x) <=> 6x-5 = 2(6x-5) =>

6x-5 = 0 => x = 5/6

--------------------------------------

5)  f,g : R --> R

a) f(x) = -2x+3 ; g(x) = x²-x+1

----

f o g = f(g(x)) = -2(x²-x+1)+3 = -2x²+2x+1

g o f = g(f(x)) = (-2x+3)²-(-2x+3)+1 = 4x²-12x+9+2x-3+1 = 4x²-10x+7

f o f = f(f(x)) = -2(-2x+3)+3 = 4x-6+3 = 4x-3

g o g = g(g(x)) = (x²-x+1)²- (x²-x+1)+1 = x⁴+x²+1-2x³-2x+2x²-x²+x-1+1 =

= x⁴-2x³+2x²-x+1

Answer 2

$\displaystyle 3.~~f,g:\mathbb{R}\rightarrow \mathbb{R},~f(x)=(f \circ f)(x),~(g \circ f)(x)=2(f \circ g)(x)\\\\ a)~f(x)=2x-1,~g(x)=3x-2\\\\ f(x)=(f\circ f)(x) \Rightarrow f(x)=f(f(x)) \Rightarrow 2x-1=2f(x)-1 \Rightarrow \\\\ \Rightarrow 2x-1=2(2x-1)-1 \Rightarrow 2x-1=4x-2-1 \Rightarrow 2x-4x=-2-1+1 \Rightarrow \\\\ \Rightarrow -2x=-2 \Rightarrow x=\frac{2}{2} \Rightarrow x=1$

$\displaystyle (g \circ f)(x)=2(f \circ g)(x) \Rightarrow g(f(x))=2f(g(x)) \Rightarrow 3f(x)-2=2 \cdot 2g(x)-1 \Rightarrow \\\\ \Rightarrow 3(2x-1)-2=2 \cdot [2(3x-2)-1 ]\Rightarrow 6x-3-2=2(6x-4-1) \Rightarrow \\\\ \Rightarrow 6x-5=2(6x-5) \Rightarrow 6x-5=12x-10 \Rightarrow 6x-12x=-10+5 \Rightarrow \\\\ \Rightarrow -6x=-5 \Rightarrow x=\frac{5}{6}$

$5.~~f,g:\mathbb{R}\rightarrow \mathbb{R}\\\\ f \circ g,~g \circ f,~f\circ f,~g \circ g=?\\\\ a)~f(x)=-2x+3,~g(x)=x^2-x+1\\\\ (f \circ g)(x)=f(g(x))=-2g(x)+3=-2\Big(x^2-x+1\Big)+3=\\\\=-2x^2+2x-2+3=-2x^2+2x+1\\\\ (g\circ f)(x)=g(f(x))=(f(x))^2-f(x)+1=(-2x+3)^2-(-2x+3)+1=\\\\=(-2x)^2 +2 \cdot (-2x) \cdot 3+3^2+2x-3+1=4x^2-12x+9+2x-3+1=\\\\=4x^2-10x+7$

$(f\circ f)(x)=f(f(x))=-2f(x)+3=-2(-2x+3)+3=\\\\=4x-6+3=4x-3\\\\ (g \circ g)(x)=g(g(x))=(g(x))^2-g(x)+1=\\\\=\Big(x^2-x+1\Big)^2-\Big(x^2-x+1\Big)+1=\\\\=\Big(x^2-x+1\Big)\Big(x^2-x+1\Big)-x^2+x-1+1=\\\\=x^4-x^3+x^2-x^3+x^2-x+x^2-x+1-x^2+x-1+1=\\\\=x^4-2x^3+2x^2-x+1$