Para as funções f(x) e g(x) dadas a seguir, determine (f \circ g)(x),(g \circ f)(x),(f \circ f)(x) \mathrm{e} (g \circ g)(x).
a) f(x)=x^{2}+3 e g(x)=\sqrt{x+1}
b) f(x)=3 x+2 e g(x)=x-2
c) f(x)=x^{2}-1 e g(x)=\frac{1}{x-1}
d) f(x)=\frac{1}{x-1} e g(x)=\sqrt{x}
e) f(x)=\frac{1}{4 x} e g(x)=\frac{1}{5 x}
f) f(x)=\log _{2} 3 x e g(x)=2^{x}
g) f(x)=3 x^{2}+\frac{5}{x} e g(x)=e^{2 x}
i) f(x)=\log x^{2} e g(x)=3 x^{3}-5 x
j) f(x)=10^{x} e g(x)=4 x^{3}-5

Passo 1

O enunciado pede para calcular composições de funções, ou seja, dado f e g, encontrar
(f\circ g)(x)=f\bigl(g(x)\bigr),\quad (g\circ f)(x)=g\bigl(f(x)\bigr),\quad (f\circ f)(x)=f\bigl(f(x)\bigr),\quad (g\circ g)(x)=g\bigl(g(x)\bigr).
Vamos aplicar essa definição em cada par (f,g) dos itens de (a) a (j).

Passo 2

A ideia geral é substituir uma função dentro da outra conforme a definição de composição. Por exemplo, para (f\circ g)(x) calculamos primeiro g(x) e em seguida aplicamos f sobre esse resultado. O mesmo vale para as demais composições.

Passo 3

Item (a): f(x)=x^{2}+3 e g(x)=\sqrt{x+1}.
(f\circ g)(x)=f\bigl(\sqrt{x+1}\bigr)=(\sqrt{x+1})^{2}+3=x+1+3=x+4,
(g\circ f)(x)=g\bigl(x^{2}+3\bigr)=\sqrt{(x^{2}+3)+1}=\sqrt{x^{2}+4},
(f\circ f)(x)=f\bigl(x^{2}+3\bigr)=(x^{2}+3)^{2}+3=x^{4}+6x^{2}+9+3=x^{4}+6x^{2}+12,
(g\circ g)(x)=g\bigl(\sqrt{x+1}\bigr)=\sqrt{\sqrt{x+1}+1}.

Passo 4

Item (b): f(x)=3x+2 e g(x)=x-2.
(f\circ g)(x)=f(x-2)=3(x-2)+2=3x-6+2=3x-4,
(g\circ f)(x)=g(3x+2)=(3x+2)-2=3x,
(f\circ f)(x)=f(3x+2)=3(3x+2)+2=9x+6+2=9x+8,
(g\circ g)(x)=g(x-2)=(x-2)-2=x-4.

Passo 5

Item (c): f(x)=x^{2}-1 e g(x)=\dfrac1{x-1}.
(f\circ g)(x)=f\Bigl(\tfrac1{x-1}\Bigr)=\Bigl(\tfrac1{x-1}\Bigr)^{2}-1=\frac1{(x-1)^{2}}-1,
(g\circ f)(x)=g(x^{2}-1)=\frac1{(x^{2}-1)-1}=\frac1{x^{2}-2},
(f\circ f)(x)=f(x^{2}-1)=(x^{2}-1)^{2}-1=x^{4}-2x^{2}+1-1=x^{4}-2x^{2},
(g\circ g)(x)=g\Bigl(\tfrac1{x-1}\Bigr)=\frac1{\bigl(\tfrac1{x-1}\bigr)-1} =\frac1{\frac{1-(x-1)}{\,x-1\,}} =\frac{x-1}{2-x}.

Passo 6

Item (d): f(x)=\dfrac1{x-1} e g(x)=\sqrt{x}.
(f\circ g)(x)=f\bigl(\sqrt{x}\bigr)=\frac1{\sqrt{x}-1},
(g\circ f)(x)=g\Bigl(\tfrac1{x-1}\Bigr)=\sqrt{\frac1{x-1}},
(f\circ f)(x)=f\Bigl(\tfrac1{x-1}\Bigr)=\frac1{\bigl(\tfrac1{x-1}\bigr)-1} =\frac{x-1}{2-x},
(g\circ g)(x)=g\bigl(\sqrt{x}\bigr)=\sqrt{\sqrt{x}}=x^{1/4}.

Passo 7

Item (e): f(x)=\dfrac1{4x} e g(x)=\dfrac1{5x}.
(f\circ g)(x)=f\Bigl(\tfrac1{5x}\Bigr)=\frac1{4\bigl(\tfrac1{5x}\bigr)}=\frac{5x}{4},
(g\circ f)(x)=g\Bigl(\tfrac1{4x}\Bigr)=\frac1{5\bigl(\tfrac1{4x}\bigr)}=\frac{4x}{5},
(f\circ f)(x)=f\Bigl(\tfrac1{4x}\Bigr)=\frac1{4\bigl(\tfrac1{4x}\bigr)}=x,
(g\circ g)(x)=g\Bigl(\tfrac1{5x}\Bigr)=\frac1{5\bigl(\tfrac1{5x}\bigr)}=x.

Passo 8

Item (f): f(x)=\log_{2}(3x) e g(x)=2^{x}.
(f\circ g)(x)=\log_{2}\bigl(3\cdot2^{x}\bigr)=\log_{2}3 + x,
(g\circ f)(x)=2^{\log_{2}(3x)}=3x,
(f\circ f)(x)=\log_{2}\bigl(3\,\log_{2}(3x)\bigr),
(g\circ g)(x)=2^{2^{x}}.

Passo 9

Item (g): f(x)=3x^{2}+\dfrac5{x} e g(x)=e^{2x}.
(f\circ g)(x)=3\bigl(e^{2x}\bigr)^{2}+\frac5{e^{2x}} =3e^{4x}+5e^{-2x},
(g\circ f)(x)=e^{2\bigl(3x^{2}+\frac5{x}\bigr)} =e^{6x^{2}+\tfrac{10}{x}},
(f\circ f)(x)=3\Bigl(3x^{2}+\frac5{x}\Bigr)^{2} +\frac5{\,3x^{2}+\frac5{x}\,},
(g\circ g)(x)=e^{2\,e^{2x}}.

Passo 10

Item (i): f(x)=\log(x^{2}) e g(x)=3x^{3}-5x.
(f\circ g)(x)=\log\bigl((3x^{3}-5x)^{2}\bigr),
(g\circ f)(x)=3\bigl(\log(x^{2})\bigr)^{3}-5\log(x^{2}),
(f\circ f)(x)=\log\Bigl(\bigl(\log(x^{2})\bigr)^{2}\Bigr),
(g\circ g)(x)=3\bigl(3x^{3}-5x\bigr)^{3}-5\bigl(3x^{3}-5x\bigr).

Passo 11

Item (j): f(x)=10^{x} e g(x)=4x^{3}-5.
(f\circ g)(x)=10^{\,4x^{3}-5},
(g\circ f)(x)=4\bigl(10^{x}\bigr)^{3}-5=4\cdot10^{3x}-5,
(f\circ f)(x)=10^{\,10^{x}},
(g\circ g)(x)=4\bigl(4x^{3}-5\bigr)^{3}-5.

Resposta

a) (f\circ g)(x)=x+4,\;(g\circ f)(x)=\sqrt{x^{2}+4},\;(f\circ f)(x)=x^{4}+6x^{2}+12,\;(g\circ g)(x)=\sqrt{\sqrt{x+1}+1}.
b) (f\circ g)(x)=3x-4,\;(g\circ f)(x)=3x,\;(f\circ f)(x)=9x+8,\;(g\circ g)(x)=x-4.
c) (f\circ g)(x)=\frac1{(x-1)^{2}}-1,\;(g\circ f)(x)=\frac1{x^{2}-2},\;(f\circ f)(x)=x^{4}-2x^{2},\;(g\circ g)(x)=\frac{x-1}{2-x}.
d) (f\circ g)(x)=\frac1{\sqrt{x}-1},\;(g\circ f)(x)=\sqrt{\frac1{x-1}},\;(f\circ f)(x)=\frac{x-1}{2-x},\;(g\circ g)(x)=x^{1/4}.
e) (f\circ g)(x)=\tfrac{5x}{4},\;(g\circ f)(x)=\tfrac{4x}{5},\;(f\circ f)(x)=x,\;(g\circ g)(x)=x.
f) (f\circ g)(x)=\log_{2}3 + x,\;(g\circ f)(x)=3x,\;(f\circ f)(x)=\log_{2}\bigl(3\log_{2}(3x)\bigr),\;(g\circ g)(x)=2^{2^{x}}.
g) (f\circ g)(x)=3e^{4x}+5e^{-2x},\;(g\circ f)(x)=e^{6x^{2}+10/x},\;(f\circ f)(x)=3\bigl(3x^{2}+\tfrac5{x}\bigr)^{2}+\tfrac5{3x^{2}+\tfrac5{x}},\;(g\circ g)(x)=e^{2e^{2x}}.
i) (f\circ g)(x)=\log\bigl((3x^{3}-5x)^{2}\bigr),\;(g\circ f)(x)=3(\log(x^{2}))^{3}-5\log(x^{2}),\;(f\circ f)(x)=\log\bigl((\log(x^{2}))^{2}\bigr),\;(g\circ g)(x)=3(3x^{3}-5x)^{3}-5(3x^{3}-5x).
j) (f\circ g)(x)=10^{4x^{3}-5},\;(g\circ f)(x)=4\cdot10^{3x}-5,\;(f\circ f)(x)=10^{10^{x}},\;(g\circ g)(x)=4(4x^{3}-5)^{3}-5.

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