Given:
The given functions are,
f(x)=2x
(1)
g(x)=xx+2
(2)
Calculation:
The function
f∘g
is written as,
(f∘g)(x)=f(g(x))
Substitute
xx+2
for
g(x)
in above expression,
(f∘g)(x)=f(g(x))
(3)
Substitute
xx+2
for x in equation (1), to find the function
f(xx+2)
,
f(xx+2)=2(xx+2)=2(x+2)x=2x+4x
Substitute
2x+4x
for
f(xx+2)
in equation (3), to find the function
f∘g
,
(f∘g)(x)=2x+4x
The required function is
(f∘g)(x)=2x+4x
.
All possible inputs for a function is called the domain of the function.
The function
(f∘g)(x)=2x+4x
defined for all real numbers lie except 0 and
−2
because at
x=−2
the denominator function
xx+2
is not defined, therefore the domain of the function is the set of all real numbers except
−2
and 0. The domain of the function is
{x|x≠−2,0}
.
The function
g∘f
is written as,
(g∘f)(x)=g(f(x))
Substitute
2x
for
f(x)
in above expression,
(g∘f)(x)=g(2x)
(4)
Substitute
2x
for x in equation (2), to find the function
g(2x)
,
g(2x)=(2x)(2x)+2=(2x)(2+2xx)=22+2x=11+x
Substitute
11+x
for
g(2x)
in equation (4), to find the function
g∘f
,
(g∘f)(x)=11+x
The required function is
(g∘f)(x)=11+x
.
All possible inputs for a function is called the domain of the function.
The function
(g∘f)(x)=11+x
is defined for all real numbers except
−1
and 0 because at
x=0
the denominator function
2x+2
is not defined, therefore the domain of the function is the set of all real numbers except
−1
and 0. The domain of the function is
{x|x≠0,−1}
.
Thus, the function
g∘f
is
(g∘f)(x)=11+x
and its domain is
{x|x≠0,−1}
.
The function
f∘f
is written as,
(f∘f)(x)=f(f(x))
Substitute
2x
for
f(x)
in above expression,
(f∘f)(x)=f(2x)
(5)
Substitute
2x
for x in equation (1), to find the function
f(2x)
,
f(2x)=2(2x)=2x2=x
Substitute x for
f(2x)
in equation (5), to find the function
f∘f
,
(f∘f)(x)=x
The required function is
(f∘f)(x)=x
.
All possible inputs for a function is called the domain of the function.
The function
(f∘f)(x)=x
is defined for all real numbers except 0 because at
x=0
the denominator function
2x
is not defined, therefore the domain of the function is the set of all real numbers except 0. The domain of the function is
{x|x≠0}
.
The function
g∘g
is written as shown below,
(g∘g)(x)=g(g(x))
Substitute
xx+2
for
g(x)
in above expression,
(g∘g)(x)=g(xx+2)
(6)
Substitute
xx+2
for x in equation (2), to find the function
g(xx+2)
,
g(xx+2)=(xx+2)(xx+2)+2=(xx+2)(x+2(x+2)x+2)=xx+2x+4=x3x+4
Substitute
x3x+4
for
g(xx+2)
in equation (6), to find the function
g∘g
,
(g∘g)(x)=x3x+4
The required function is
(g∘g)(x)=x3x+4
.
All possible inputs for a function is called the domain of the function.
The function
(g∘g)(x)=x3x+4
defined for all real numbers except
−43
and
−2
because at
x=−2
the denominator function
(xx+2+2)
is not defined, therefore the domain of the function is the set of all real numbers except
−43
and
−2
. The domain of the function is
{x|x≠−43,−2}
.