If g(x) = log, xạ, then range of the function g[f(x)]
is
Answers
Answer:
Answer
f(x)=
1−x
For domain, 1−x≥0
⇒ x≤1
⇒ Domain of f=(−∞,1]
⇒ f:(∞,1]→(0,∞)
g(x)=log
e
x
Clearly, the range of g is not a subset of the domain of f.
So, we need to compute the domain of f∘g.
⇒ Domain (f∘g)=(0,e)→R
⇒ (f∘g)(x)=f[g(x)]
=f(log
e
x)
=
1−log
e
x
The range of f is a subset of the domain of g.
⇒ g∘f:(−∞,1]→R
⇒ (g∘f)(x)=g[f(x)]
=g(
1−x
)
=log
e
(1−x)
2
1
=
2
1
log
e
(1−x)
f(x)=
1−x
For domain, 1−x≥0
⇒ x≤1
⇒ Domain of f=(−∞,1]
⇒ f:(∞,1]→(0,∞)
g(x)=log
e
x
Clearly, the range of g is not a subset of the domain of f.
So, we need to compute the domain of f∘g.
⇒ Domain (f∘g)=(0,e)→R
⇒ (f∘g)(x)=f[g(x)]
=f(log
e
x)
=
1−log
e
x
The range of f is a subset of the domain of g.
⇒ g∘f:(−∞,1]→R
⇒ (g∘f)(x)=g[f(x)]
=g(
1−x
)
=log
e
(1−x)
2
1
=
2
1
log
e
(1−x)