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Answer :
To find the composition of the functions [tex]\( g \)[/tex] and [tex]\( f \)[/tex], denoted as [tex]\( g \circ f(x) = g(f(x)) \)[/tex], we need to follow these steps:
1. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
- First, we have the function [tex]\( f(x) = 3x^2 \)[/tex].
- Next, we have [tex]\( g(x) = 2x^3 \)[/tex].
- We substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(3x^2)
\][/tex]
2. Evaluate [tex]\( g(3x^2) \)[/tex]:
- The function [tex]\( g(z) = 2z^3 \)[/tex] means you need to replace [tex]\( z \)[/tex] with [tex]\( 3x^2 \)[/tex]:
[tex]\[
g(3x^2) = 2(3x^2)^3
\][/tex]
3. Simplify the expression:
- First, calculate [tex]\( (3x^2)^3 \)[/tex]:
[tex]\[
(3x^2)^3 = 3^3 \times (x^2)^3 = 27x^6
\][/tex]
- Now plug this back into the function:
[tex]\[
g(3x^2) = 2 \times 27x^6 = 54x^6
\][/tex]
So, the composition [tex]\( g \circ f(x) \)[/tex] simplifies to [tex]\( 54x^6 \)[/tex]. Thus, the answer is [tex]\( 54x^6 \)[/tex].
1. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
- First, we have the function [tex]\( f(x) = 3x^2 \)[/tex].
- Next, we have [tex]\( g(x) = 2x^3 \)[/tex].
- We substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(3x^2)
\][/tex]
2. Evaluate [tex]\( g(3x^2) \)[/tex]:
- The function [tex]\( g(z) = 2z^3 \)[/tex] means you need to replace [tex]\( z \)[/tex] with [tex]\( 3x^2 \)[/tex]:
[tex]\[
g(3x^2) = 2(3x^2)^3
\][/tex]
3. Simplify the expression:
- First, calculate [tex]\( (3x^2)^3 \)[/tex]:
[tex]\[
(3x^2)^3 = 3^3 \times (x^2)^3 = 27x^6
\][/tex]
- Now plug this back into the function:
[tex]\[
g(3x^2) = 2 \times 27x^6 = 54x^6
\][/tex]
So, the composition [tex]\( g \circ f(x) \)[/tex] simplifies to [tex]\( 54x^6 \)[/tex]. Thus, the answer is [tex]\( 54x^6 \)[/tex].
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