Answer :
To determine which pair of functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] satisfies the condition [tex]\( f \circ g = g \circ f \)[/tex], we will check each option step by step.
### Option A
Functions:
[tex]\[ f(x) = x + 3 \][/tex]
[tex]\[ g(x) = 3x \][/tex]
Compose f(g(x)):
[tex]\[ f(g(x)) = f(3x) = 3x + 3 \][/tex]
Compose g(f(x)):
[tex]\[ g(f(x)) = g(x + 3) = 3(x + 3) = 3x + 9 \][/tex]
Clearly, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] are not equal because:
[tex]\[ 3x + 3 \neq 3x + 9 \][/tex]
So, option A does not satisfy the condition [tex]\( f \circ g = g \circ f \)[/tex].
### Option B
Functions:
[tex]\[ f(x) = x + 3 \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
Compose f(g(x)):
[tex]\[ f(g(x)) = f(x - 3) = (x - 3) + 3 = x \][/tex]
Compose g(f(x)):
[tex]\[ g(f(x)) = g(x + 3) = (x + 3) - 3 = x \][/tex]
In this case, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] are equal for all [tex]\( x \)[/tex]:
[tex]\[ x = x \][/tex]
So, option B satisfies the condition [tex]\( f \circ g = g \circ f \)[/tex].
### Option C
Functions:
[tex]\[ f(x) = 3x \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
Compose f(g(x)):
[tex]\[ f(g(x)) = f(x - 3) = 3(x - 3) = 3x - 9 \][/tex]
Compose g(f(x)):
[tex]\[ g(f(x)) = g(3x) = 3x - 3 \][/tex]
Clearly, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] are not equal because:
[tex]\[ 3x - 9 \neq 3x - 3 \][/tex]
So, option C does not satisfy the condition [tex]\( f \circ g = g \circ f \)[/tex].
### Option D
Functions:
[tex]\[ f(x) = 2x + 4 \][/tex]
[tex]\[ g(x) = 2x - 4 \][/tex]
Compose f(g(x)):
[tex]\[ f(g(x)) = f(2x - 4) = 2(2x - 4) + 4 = 4x - 8 + 4 = 4x - 4 \][/tex]
Compose g(f(x)):
[tex]\[ g(f(x)) = g(2x + 4) = 2(2x + 4) - 4 = 4x + 8 - 4 = 4x + 4 \][/tex]
Clearly, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] are not equal because:
[tex]\[ 4x - 4 \neq 4x + 4 \][/tex]
So, option D does not satisfy the condition [tex]\( f \circ g = g \circ f \)[/tex].
### Conclusion
The correct answer is:
B. [tex]\( f(x) = x + 3 \)[/tex] and [tex]\( g(x) = x - 3 \)[/tex]
### Option A
Functions:
[tex]\[ f(x) = x + 3 \][/tex]
[tex]\[ g(x) = 3x \][/tex]
Compose f(g(x)):
[tex]\[ f(g(x)) = f(3x) = 3x + 3 \][/tex]
Compose g(f(x)):
[tex]\[ g(f(x)) = g(x + 3) = 3(x + 3) = 3x + 9 \][/tex]
Clearly, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] are not equal because:
[tex]\[ 3x + 3 \neq 3x + 9 \][/tex]
So, option A does not satisfy the condition [tex]\( f \circ g = g \circ f \)[/tex].
### Option B
Functions:
[tex]\[ f(x) = x + 3 \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
Compose f(g(x)):
[tex]\[ f(g(x)) = f(x - 3) = (x - 3) + 3 = x \][/tex]
Compose g(f(x)):
[tex]\[ g(f(x)) = g(x + 3) = (x + 3) - 3 = x \][/tex]
In this case, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] are equal for all [tex]\( x \)[/tex]:
[tex]\[ x = x \][/tex]
So, option B satisfies the condition [tex]\( f \circ g = g \circ f \)[/tex].
### Option C
Functions:
[tex]\[ f(x) = 3x \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
Compose f(g(x)):
[tex]\[ f(g(x)) = f(x - 3) = 3(x - 3) = 3x - 9 \][/tex]
Compose g(f(x)):
[tex]\[ g(f(x)) = g(3x) = 3x - 3 \][/tex]
Clearly, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] are not equal because:
[tex]\[ 3x - 9 \neq 3x - 3 \][/tex]
So, option C does not satisfy the condition [tex]\( f \circ g = g \circ f \)[/tex].
### Option D
Functions:
[tex]\[ f(x) = 2x + 4 \][/tex]
[tex]\[ g(x) = 2x - 4 \][/tex]
Compose f(g(x)):
[tex]\[ f(g(x)) = f(2x - 4) = 2(2x - 4) + 4 = 4x - 8 + 4 = 4x - 4 \][/tex]
Compose g(f(x)):
[tex]\[ g(f(x)) = g(2x + 4) = 2(2x + 4) - 4 = 4x + 8 - 4 = 4x + 4 \][/tex]
Clearly, [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] are not equal because:
[tex]\[ 4x - 4 \neq 4x + 4 \][/tex]
So, option D does not satisfy the condition [tex]\( f \circ g = g \circ f \)[/tex].
### Conclusion
The correct answer is:
B. [tex]\( f(x) = x + 3 \)[/tex] and [tex]\( g(x) = x - 3 \)[/tex]
