Answer :
To perform the composition of functions [tex]\( g(f(x)) \)[/tex], we'll follow these steps:
1. Identify the functions:
- [tex]\( f(x) = x^2 + 2 \)[/tex]
- [tex]\( g(x) = 4x - 1 \)[/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
- We need to find [tex]\( g(f(x)) \)[/tex]. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex], where [tex]\( f(x) \)[/tex] is [tex]\( x^2 + 2 \)[/tex].
- So, [tex]\( g(f(x)) = g(x^2 + 2) \)[/tex].
3. Evaluate [tex]\( g(x^2 + 2) \)[/tex]:
- [tex]\( g(x) = 4x - 1 \)[/tex], therefore [tex]\( g(x^2 + 2) = 4(x^2 + 2) - 1 \)[/tex].
4. Simplify the expression:
- [tex]\( 4(x^2 + 2) - 1 \)[/tex]
- Distribute the 4: [tex]\( 4 \cdot x^2 + 4 \cdot 2 - 1 \)[/tex]
- [tex]\( 4x^2 + 8 - 1 \)[/tex]
- Combine like terms: [tex]\( 4x^2 + 7 \)[/tex]
So, the composition [tex]\( g(f(x)) \)[/tex] simplifies to [tex]\( 4x^2 + 7 \)[/tex].
### Check the Given Choices:
- [tex]\( 4x^2 + 1 \)[/tex]: Incorrect.
- [tex]\( 4x^2 + 7 \)[/tex]: Correct.
- [tex]\( 16x^2 - 8x + 3 \)[/tex]: Incorrect.
- [tex]\( 16x^2 + 3 \)[/tex]: Incorrect.
### Domain Restriction:
The domain of [tex]\( f(x) \)[/tex] (a polynomial) is all real numbers, [tex]\(\mathbb{R}\)[/tex].
The domain of [tex]\( g(x) \)[/tex] (a linear function) is also all real numbers, [tex]\(\mathbb{R}\)[/tex].
Thus, the domain of [tex]\( g(f(x)) \)[/tex] is all real numbers, [tex]\(\mathbb{R}\)[/tex], since both component functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have domains that include all real numbers.
Conclusion: The correct simplification of [tex]\( g(f(x)) \)[/tex] is [tex]\( 4x^2 + 7 \)[/tex], and the domain of this composed function is all real numbers, [tex]\(\mathbb{R}\)[/tex].
1. Identify the functions:
- [tex]\( f(x) = x^2 + 2 \)[/tex]
- [tex]\( g(x) = 4x - 1 \)[/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
- We need to find [tex]\( g(f(x)) \)[/tex]. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex], where [tex]\( f(x) \)[/tex] is [tex]\( x^2 + 2 \)[/tex].
- So, [tex]\( g(f(x)) = g(x^2 + 2) \)[/tex].
3. Evaluate [tex]\( g(x^2 + 2) \)[/tex]:
- [tex]\( g(x) = 4x - 1 \)[/tex], therefore [tex]\( g(x^2 + 2) = 4(x^2 + 2) - 1 \)[/tex].
4. Simplify the expression:
- [tex]\( 4(x^2 + 2) - 1 \)[/tex]
- Distribute the 4: [tex]\( 4 \cdot x^2 + 4 \cdot 2 - 1 \)[/tex]
- [tex]\( 4x^2 + 8 - 1 \)[/tex]
- Combine like terms: [tex]\( 4x^2 + 7 \)[/tex]
So, the composition [tex]\( g(f(x)) \)[/tex] simplifies to [tex]\( 4x^2 + 7 \)[/tex].
### Check the Given Choices:
- [tex]\( 4x^2 + 1 \)[/tex]: Incorrect.
- [tex]\( 4x^2 + 7 \)[/tex]: Correct.
- [tex]\( 16x^2 - 8x + 3 \)[/tex]: Incorrect.
- [tex]\( 16x^2 + 3 \)[/tex]: Incorrect.
### Domain Restriction:
The domain of [tex]\( f(x) \)[/tex] (a polynomial) is all real numbers, [tex]\(\mathbb{R}\)[/tex].
The domain of [tex]\( g(x) \)[/tex] (a linear function) is also all real numbers, [tex]\(\mathbb{R}\)[/tex].
Thus, the domain of [tex]\( g(f(x)) \)[/tex] is all real numbers, [tex]\(\mathbb{R}\)[/tex], since both component functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have domains that include all real numbers.
Conclusion: The correct simplification of [tex]\( g(f(x)) \)[/tex] is [tex]\( 4x^2 + 7 \)[/tex], and the domain of this composed function is all real numbers, [tex]\(\mathbb{R}\)[/tex].
