To find [tex]\((f-g)(x)\)[/tex], we need to subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]. Let's go through the steps together:
1. Identify the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = 7x^2 - x - 1
\][/tex]
[tex]\[
g(x) = 2x + 1
\][/tex]
2. Express [tex]\((f-g)(x)\)[/tex]:
[tex]\[
(f-g)(x) = f(x) - g(x)
\][/tex]
3. Substitute the given functions into the expression:
[tex]\[
(f-g)(x) = (7x^2 - x - 1) - (2x + 1)
\][/tex]
4. Distribute the negative sign through the terms of [tex]\(g(x)\)[/tex]:
[tex]\[
(f-g)(x) = 7x^2 - x - 1 - 2x - 1
\][/tex]
5. Combine like terms:
- The [tex]\(x^2\)[/tex] term remains [tex]\(7x^2\)[/tex].
- The [tex]\(x\)[/tex] terms combine: [tex]\(-x - 2x = -3x\)[/tex].
- The constant terms combine: [tex]\(-1 - 1 = -2\)[/tex].
6. Write the simplified expression:
[tex]\[
(f-g)(x) = 7x^2 - 3x - 2
\][/tex]
Therefore, the result is:
[tex]\[
(f-g)(x) = 7x^2 - 3x - 2
\][/tex]
