Question

Dadas as funções f(x)=x²-5,g(x)=x+3 e h(x)=8x,calcule:
a) gof e fog
b) gof + fog+f(h(x))

Answer 1

$f(x)=x^2-5$

$g(x)=x+3$

a) $g\text{o}f=g(f(x))$

$g(f(x))=g(x^2-5)=(x^2-5)+3=x^2-2$.

$f\text{o}\text{g}=f(g(x))$

$f(x+3)=(x+3)^2-5=x^2+6x+9-5=x^2+6x+4$

b) $h(x)=8x$

$f(x)=x^2-5$

$f(h(x))=f(8x)=(8x)^2-5=64x^2-5$

$g\text{o}f=x^2-2$

$f\text{o}g=x^2+6x+4$

$g\text{o}f+f\text{o}g+f(h(x))=x^2-2+x^2+6x+4+64x^2-5=66x^2+6x-3$

Answer 2

a)

$gof=(x^2-5)+3=x^2-2\\ \\ fog=(x+3)^2-5=x^2+6x+9-5=x^2+6x+4$

b)
$f(h(x))=(8x)^2-5=64x^2-5\\ \\ gof+fog+f(h(x))=\\ \\ x^2-2+x^2+6x+4+64x^2-5=66x^2+6x-3$