To find [tex]\((f - g)(x)\)[/tex] given the functions [tex]\(f(x) = -2x + 4\)[/tex] and [tex]\(g(x) = x^2 - 3x + 5\)[/tex], follow these steps:
1. Express [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = -2x + 4
\][/tex]
[tex]\[
g(x) = x^2 - 3x + 5
\][/tex]
2. Define [tex]\((f - g)(x)\)[/tex] as the difference between [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/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]\[
(f - g)(x) = (-2x + 4) - (x^2 - 3x + 5)
\][/tex]
4. Distribute the negative sign through the second term:
[tex]\[
(f - g)(x) = -2x + 4 - x^2 + 3x - 5
\][/tex]
5. Combine like terms:
[tex]\[
-x^2 + (-2x + 3x) + (4 - 5)
\][/tex]
[tex]\[
-x^2 + x - 1
\][/tex]
Therefore, the correct expression for [tex]\((f - g)(x)\)[/tex] is:
[tex]\[
(f - g)(x) = -x^2 + x - 1
\][/tex]
So, the correct answer is:
[tex]\((f - g)(x) = -x^2 + x - 1\)[/tex].
