Question

Considere as funções F e G de R em R tais que f(x)=3x+1 e g(x)=x-2
Determine:
a) g[f(1)]
b) g[f(2)]
c) g[f(x)]
d) f[g(1)]
e) f[g(2)]
f) f[g(x)]

Answer 1

$f(x)=3x+1~~\text{ e }~~g(x)=x-2.$

a) $g(f(1)):$

$f(1)=3\cdot 1+1=4\\\\\\ \therefore~~g[f(1)]=g(4)\\\\ g[f(1)]=4-2\\\\ \boxed{\begin{array}{c}g[f(1)]=2 \end{array}}$


b) $g[f(2)]:$

$f(2)=3\cdot 2+1=7\\\\\\ \therefore~~g[f(2)]=g(7)\\\\ g[f(2)]=7-2\\\\ \boxed{\begin{array}{c}g[f(2)]=5 \end{array}}$


c) Se $g(x)=x-2\,,$ então

$g[f(x)]=f(x)-2\\\\ g[f(x)]=(3x+1)-2\\\\ g[f(x)]=3x+1-2\\\\ \boxed{\begin{array}{c}g[f(x)]=3x-1 \end{array}}$


d) $f[g(1)]:$

$g(1)=1-2=-1\\\\\\ \therefore~~f[g(1)]=f(-1)\\\\ f[g(1)]=3\cdot (-1)+1\\\\ f[g(1)]=-3+1\\\\ \boxed{\begin{array}{c}f[g(1)]=-2 \end{array}}$


e) $f[g(2)]:$

$g(2)=2-2=0\\\\\\ \therefore~~f[g(2)]=f(0)\\\\ f[g(2)]=3\cdot 0+1\\\\ f[g(2)]=0+1\\\\ \boxed{\begin{array}{c}f[g(2)]=1 \end{array}}$


f) Se $f(x)=3x+1\,,$ então

$f[g(x)]=3\,g(x)+1\\\\ f[g(x)]=3\,(x-2)+1\\\\ f[g(x)]=3x-6+1\\\\ \boxed{\begin{array}{c}f[g(x)]=3x-5 \end{array}}$