Answer :
To find [tex]\( g \circ f(x) = g(f(x)) \)[/tex], you'll need to substitute the output of the function [tex]\( f(x) \)[/tex] into the function [tex]\( g(x) \)[/tex]. Here are the steps:
1. Identify the given functions:
- [tex]\( f(x) = 3x^2 \)[/tex]
- [tex]\( g(x) = 2x^3 \)[/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
- Since [tex]\( f(x) = 3x^2 \)[/tex], you'll substitute [tex]\( 3x^2 \)[/tex] into [tex]\( g(x) \)[/tex].
- So, [tex]\( g(f(x)) = g(3x^2) \)[/tex].
3. Evaluate [tex]\( g(3x^2) \)[/tex]:
- Recall that [tex]\( g(x) = 2x^3 \)[/tex].
- Replace [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] with [tex]\( 3x^2 \)[/tex]:
[tex]\[ g(3x^2) = 2(3x^2)^3 \][/tex]
4. Simplify the expression:
- First calculate [tex]\( (3x^2)^3 \)[/tex]:
[tex]\((3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6\)[/tex]
- Then multiply by 2:
[tex]\( 2 \cdot 27x^6 = 54x^6 \)[/tex]
The result is:
[tex]\[ g(f(x)) = 54x^6 \][/tex]
Therefore, [tex]\( g \circ f(x) = 54x^6 \)[/tex].
1. Identify the given functions:
- [tex]\( f(x) = 3x^2 \)[/tex]
- [tex]\( g(x) = 2x^3 \)[/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
- Since [tex]\( f(x) = 3x^2 \)[/tex], you'll substitute [tex]\( 3x^2 \)[/tex] into [tex]\( g(x) \)[/tex].
- So, [tex]\( g(f(x)) = g(3x^2) \)[/tex].
3. Evaluate [tex]\( g(3x^2) \)[/tex]:
- Recall that [tex]\( g(x) = 2x^3 \)[/tex].
- Replace [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] with [tex]\( 3x^2 \)[/tex]:
[tex]\[ g(3x^2) = 2(3x^2)^3 \][/tex]
4. Simplify the expression:
- First calculate [tex]\( (3x^2)^3 \)[/tex]:
[tex]\((3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6\)[/tex]
- Then multiply by 2:
[tex]\( 2 \cdot 27x^6 = 54x^6 \)[/tex]
The result is:
[tex]\[ g(f(x)) = 54x^6 \][/tex]
Therefore, [tex]\( g \circ f(x) = 54x^6 \)[/tex].
