Determine
e
, se:
a)
b)
c)
d)
e)
f)
g)
a)
b)
c)
d)
e)
f)
g)
Passo 1
O enunciado pede que, para cada par de funções f e g, a gente calcule as operações soma, diferença, produto e quociente de funções. Vamos usar as definições:
(f+g)(x)=f(x)+g(x),
(f−g)(x)=f(x)−g(x),
(f·g)(x)=f(x)·g(x),
(f/g)(x)=f(x)/g(x).
(f+g)(x)=f(x)+g(x),
(f−g)(x)=f(x)−g(x),
(f·g)(x)=f(x)·g(x),
(f/g)(x)=f(x)/g(x).
Passo 2
Para a) f(x)=2x e g(x)=x²+2, aplicando as definições temos:
(f+g)(x)=2x+(x²+2)=x²+2x+2,
(f−g)(x)=2x−(x²+2)=−x²+2x−2,
(f·g)(x)=2x·(x²+2)=2x³+4x,
(f/g)(x)=2x/(x²+2).
(f+g)(x)=2x+(x²+2)=x²+2x+2,
(f−g)(x)=2x−(x²+2)=−x²+2x−2,
(f·g)(x)=2x·(x²+2)=2x³+4x,
(f/g)(x)=2x/(x²+2).
Passo 3
Para b) f(x)=3x−2 e g(x)=|x+2|, usando as definições:
(f+g)(x)=3x−2+|x+2|,
(f−g)(x)=3x−2−|x+2|,
(f·g)(x)=(3x−2)·|x+2|,
(f/g)(x)=(3x−2)/|x+2|.
(f+g)(x)=3x−2+|x+2|,
(f−g)(x)=3x−2−|x+2|,
(f·g)(x)=(3x−2)·|x+2|,
(f/g)(x)=(3x−2)/|x+2|.
Passo 4
Para c) f(x)=√(x+1) e g(x)=x²−1, calculamos:
(f+g)(x)=√(x+1)+(x²−1),
(f−g)(x)=√(x+1)−(x²−1)=√(x+1)−x²+1,
(f·g)(x)=(x²−1)·√(x+1),
(f/g)(x)=√(x+1)/(x²−1).
(f+g)(x)=√(x+1)+(x²−1),
(f−g)(x)=√(x+1)−(x²−1)=√(x+1)−x²+1,
(f·g)(x)=(x²−1)·√(x+1),
(f/g)(x)=√(x+1)/(x²−1).
Passo 5
Para d) f(x)=√(x+1) e g(x)=√(x+3), obtemos:
(f+g)(x)=√(x+1)+√(x+3),
(f−g)(x)=√(x+1)−√(x+3),
(f·g)(x)=√((x+1)(x+3)),
(f/g)(x)=√(x+1)/√(x+3)=√((x+1)/(x+3)).
(f+g)(x)=√(x+1)+√(x+3),
(f−g)(x)=√(x+1)−√(x+3),
(f·g)(x)=√((x+1)(x+3)),
(f/g)(x)=√(x+1)/√(x+3)=√((x+1)/(x+3)).
Passo 6
Para e) f(x)=x⁴ e g(x)=(1/x)⁴=1/x⁴, resulta:
(f+g)(x)=x⁴+1/x⁴,
(f−g)(x)=x⁴−1/x⁴,
(f·g)(x)=x⁴·(1/x⁴)=1,
(f/g)(x)=x⁴/(1/x⁴)=x⁸.
(f+g)(x)=x⁴+1/x⁴,
(f−g)(x)=x⁴−1/x⁴,
(f·g)(x)=x⁴·(1/x⁴)=1,
(f/g)(x)=x⁴/(1/x⁴)=x⁸.
Passo 7
Para f) f(x)=1/x e g(x)=x², então:
(f+g)(x)=1/x+x²,
(f−g)(x)=1/x−x²,
(f·g)(x)=(1/x)·x²=x,
(f/g)(x)=(1/x)/x²=1/x³.
(f+g)(x)=1/x+x²,
(f−g)(x)=1/x−x²,
(f·g)(x)=(1/x)·x²=x,
(f/g)(x)=(1/x)/x²=1/x³.
Passo 8
Para g) f(x)=x³+x² e g(x)=(1/x²)⁴=1/x⁸, temos:
(f+g)(x)=x³+x²+1/x⁸,
(f−g)(x)=x³+x²−1/x⁸,
(f·g)(x)=(x³+x²)·(1/x⁸)=x³/x⁸+x²/x⁸=1/x⁵+1/x⁶=
(x+1)/x⁶,
(f/g)(x)=(x³+x²)/(1/x⁸)=x¹¹+x¹⁰.
(f+g)(x)=x³+x²+1/x⁸,
(f−g)(x)=x³+x²−1/x⁸,
(f·g)(x)=(x³+x²)·(1/x⁸)=x³/x⁸+x²/x⁸=1/x⁵+1/x⁶=
(x+1)/x⁶,
(f/g)(x)=(x³+x²)/(1/x⁸)=x¹¹+x¹⁰.
Resposta
a) (f+g)(x)=x²+2x+2, (f−g)(x)=−x²+2x−2, (f·g)(x)=2x³+4x, (f/g)(x)=2x/(x²+2)
b) (f+g)(x)=3x−2+|x+2|, (f−g)(x)=3x−2−|x+2|, (f·g)(x)=(3x−2)|x+2|, (f/g)(x)=(3x−2)/|x+2|
c) (f+g)(x)=√(x+1)+(x²−1), (f−g)(x)=√(x+1)−x²+1, (f·g)(x)=(x²−1)√(x+1), (f/g)(x)=√(x+1)/(x²−1)
d) (f+g)(x)=√(x+1)+√(x+3), (f−g)(x)=√(x+1)−√(x+3), (f·g)(x)=√((x+1)(x+3)), (f/g)(x)=√((x+1)/(x+3))
e) (f+g)(x)=x⁴+1/x⁴, (f−g)(x)=x⁴−1/x⁴, (f·g)(x)=1, (f/g)(x)=x⁸
f) (f+g)(x)=1/x+x², (f−g)(x)=1/x−x², (f·g)(x)=x, (f/g)(x)=1/x³
g) (f+g)(x)=x³+x²+1/x⁸, (f−g)(x)=x³+x²−1/x⁸, (f·g)(x)=(x+1)/x⁶, (f/g)(x)=x¹¹+x¹⁰
b) (f+g)(x)=3x−2+|x+2|, (f−g)(x)=3x−2−|x+2|, (f·g)(x)=(3x−2)|x+2|, (f/g)(x)=(3x−2)/|x+2|
c) (f+g)(x)=√(x+1)+(x²−1), (f−g)(x)=√(x+1)−x²+1, (f·g)(x)=(x²−1)√(x+1), (f/g)(x)=√(x+1)/(x²−1)
d) (f+g)(x)=√(x+1)+√(x+3), (f−g)(x)=√(x+1)−√(x+3), (f·g)(x)=√((x+1)(x+3)), (f/g)(x)=√((x+1)/(x+3))
e) (f+g)(x)=x⁴+1/x⁴, (f−g)(x)=x⁴−1/x⁴, (f·g)(x)=1, (f/g)(x)=x⁸
f) (f+g)(x)=1/x+x², (f−g)(x)=1/x−x², (f·g)(x)=x, (f/g)(x)=1/x³
g) (f+g)(x)=x³+x²+1/x⁸, (f−g)(x)=x³+x²−1/x⁸, (f·g)(x)=(x+1)/x⁶, (f/g)(x)=x¹¹+x¹⁰