Question

Se considera functiile liniare f,g :R cu valori in R ce satisfac relatiile:
f(x-1)=3x-2+g(1) -3f(1) si g(x-1)=x +2+2g(1) -f(1), oricare ar fi x apartine lui R
a)Sa se afle f(1) si g(1)
b)Sa se afle punctul de intersectie al graficelor functiilor f si g
c)Sa se calculeze aria figurii determinate de graficele celor doua functii si axa OY.​

Answer 1

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

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

a) Pentru x = 2:

f(2-1) = 6-2+g(1)-3f(1)

g(2-1) = 4+2g(1)-3f(1)

f(1) = 4+g(1)-3f(1)

g(1) = 4+2g(1)-3f(1)

4f(1)-g(1) = 4

3f(1)-g(1) = 4

------------------ (-)

f(1) = 0 => g(1) = -4

=> f(x-1) = 3x-2-4 = 3x-6

=> g(x-1) = x+2-8 = x-6

b) f(x-1) = 3x-6

x-1 = t => x = t+1

=> f(t) = 3(t+1)-6 = 3t-3 => f(x) = 3x-3

g(x-1) = x-6

x-1 = t => x = t+1

g(t) = t+1-6 = t-5 => g(x) = x-5

f(x) = g(x) => 3x-3 = x-5 => 2x = -2 => x = -1

=> f(-1) = g(-1) = -6

=> Punctul de intersectie este (-1,-6).