Question

Sejam as funções reais f e g tais que:
fog(x)=|4x2 - 4x - 3| e g(x)=20-1. Qual o
valor de f(-1)?

Answer 1

Explicação passo-a-passo:

Seja $\sf f(x)=|~ax^2+bx+c~|$

Temos que:

$\sf f(g(x))=f(2x-1)$

$\sf f(g(x))=|~a\cdot(2x-1)^2+b\cdot(2x-1)+c~|$

$\sf f(g(x))=|~a\cdot(4x^2-4x+1)+b\cdot(2x-1)+c~|$

$\sf f(g(x))=|~4ax^2-4ax+4a+2bx-b+c~|$

$\sf f(g(x))=|~4ax^2+2bx-4ax+4a-b+c~|$

$\sf f(g(x))=|~4ax^2+(2b-4a)x+4a-b+c~|$

Assim:

$\sf |~4ax^2+(2b-4a)x+4a-b+c~|=|~4x^2-4x-3~|$

Devemos ter:

• $\sf 4a=4$

$\sf a=\dfrac{4}{4}$

$\sf \red{a=1}$

• $\sf 2b-4a=-4$

$\sf 2b-4\cdot1=-4$

$\sf 2b-4=-4$

$\sf 2b=-4+4$

$\sf 2b=0$

$\sf b=\dfrac{0}{2}$

$\sf \red{b=0}$

• $\sf 4a-b+c=-3$

$\sf 4\cdot1-0+c=-3$

$\sf 4-0+c=-3$

$\sf 4+c=-3$

$\sf c=-3-4$

$\sf \red{c=-7}$

Desse modo:

$\sf f(x)=|~x^2-7~|$

Logo:

$\sf f(-1)=|~(-1)^2-7~|$

$\sf f(-1)=|~1-7~|$

$\sf f(-1)=|~-6~|$

$\sf \red{f(-1)=6}$