To find [tex]\((f - g)(x)\)[/tex] where [tex]\(f(x) = 2x^2 + 1\)[/tex] and [tex]\(g(x) = x^2 - 7\)[/tex], we need to subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex].
Step-by-Step solution:
1. Write the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 2x^2 + 1
\][/tex]
[tex]\[
g(x) = x^2 - 7
\][/tex]
2. Form the expression for [tex]\((f - g)(x)\)[/tex]:
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
3. Substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] into the equation:
[tex]\[
(f - g)(x) = (2x^2 + 1) - (x^2 - 7)
\][/tex]
4. Distribute the negative sign through the second expression:
[tex]\[
(f - g)(x) = 2x^2 + 1 - x^2 + 7
\][/tex]
5. Combine like terms:
[tex]\[
(f - g)(x) = (2x^2 - x^2) + (1 + 7)
\][/tex]
[tex]\[
(f - g)(x) = x^2 + 8
\][/tex]
After simplifying, we find that [tex]\((f - g)(x)\)[/tex] is:
[tex]\[
(f - g)(x) = 3x^2 + 8
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{3x^2 + 8}
\][/tex]
Thus, the choice is:
[tex]\[
\text{D. } 3x^2 + 8
\][/tex]
