Answer :
To find [tex]\( g \circ f(x) = g(f(x)) \)[/tex], we need to substitute the expression for [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]. Let's break it down step-by-step:
1. Define the Functions:
- [tex]\( f(x) = 3x^2 \)[/tex]
- [tex]\( g(x) = 2x^3 \)[/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex] to find [tex]\( g(f(x)) \)[/tex]:
- First, find [tex]\( f(x) \)[/tex], which is [tex]\( 3x^2 \)[/tex].
- Now, replace [tex]\( x \)[/tex] in [tex]\( g(x) = 2x^3 \)[/tex] with [tex]\( f(x) = 3x^2 \)[/tex].
3. Calculation of [tex]\( g(f(x)) \)[/tex]:
- Substitute: [tex]\( g(f(x)) = g(3x^2) = 2(3x^2)^3 \)[/tex].
4. Simplify:
- Calculate [tex]\( (3x^2)^3 \)[/tex]:
[tex]\[
(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6
\][/tex]
- Now, multiply this result by 2 because [tex]\( g(x) = 2x^3 \)[/tex]:
[tex]\[
g(3x^2) = 2 \cdot 27x^6 = 54x^6
\][/tex]
Therefore, the function [tex]\( g \circ f(x) \)[/tex] results in [tex]\( 54x^6 \)[/tex], which matches one of the options provided. The correct answer is [tex]\( 54x^6 \)[/tex].
1. Define the Functions:
- [tex]\( f(x) = 3x^2 \)[/tex]
- [tex]\( g(x) = 2x^3 \)[/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex] to find [tex]\( g(f(x)) \)[/tex]:
- First, find [tex]\( f(x) \)[/tex], which is [tex]\( 3x^2 \)[/tex].
- Now, replace [tex]\( x \)[/tex] in [tex]\( g(x) = 2x^3 \)[/tex] with [tex]\( f(x) = 3x^2 \)[/tex].
3. Calculation of [tex]\( g(f(x)) \)[/tex]:
- Substitute: [tex]\( g(f(x)) = g(3x^2) = 2(3x^2)^3 \)[/tex].
4. Simplify:
- Calculate [tex]\( (3x^2)^3 \)[/tex]:
[tex]\[
(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6
\][/tex]
- Now, multiply this result by 2 because [tex]\( g(x) = 2x^3 \)[/tex]:
[tex]\[
g(3x^2) = 2 \cdot 27x^6 = 54x^6
\][/tex]
Therefore, the function [tex]\( g \circ f(x) \)[/tex] results in [tex]\( 54x^6 \)[/tex], which matches one of the options provided. The correct answer is [tex]\( 54x^6 \)[/tex].
