To solve the problem, we first need to determine the function [tex]\( f(x) \)[/tex] given that [tex]\( f(x-2) = 2x^2 - 3x + 4 \)[/tex].
1. Identify the expression for [tex]\( f(x) \)[/tex]:
Given that [tex]\( f(x-2) = 2x^2 - 3x + 4 \)[/tex], we need to find the expression for [tex]\( f(x) \)[/tex].
Let's substitute [tex]\( x-2 \)[/tex] with [tex]\( t \)[/tex]:
[tex]\[
f(t) = 2(x)^2 - 3(x) + 4
\][/tex]
We replace [tex]\( x \)[/tex] with [tex]\( (x+2) \)[/tex]:
[tex]\[
f(x) = 2(x + 2)^2 - 3(x + 2) + 4
\][/tex]
2. Expand and simplify the expression:
[tex]\[
f(x) = 2(x+2)^2 - 3(x+2) + 4
\][/tex]
Let's expand [tex]\( (x+2)^2 \)[/tex]:
[tex]\[
(x+2)^2 = x^2 + 4x + 4
\][/tex]
Substitute back into the function:
[tex]\[
f(x) = 2(x^2 + 4x + 4) - 3(x + 2) + 4
\][/tex]
[tex]\[
f(x) = 2x^2 + 8x + 8 - 3x - 6 + 4
\][/tex]
Combine like terms:
[tex]\[
f(x) = 2x^2 + 5x + 6
\][/tex]
3. Find the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( (x-1) \)[/tex]:
According to the Remainder Theorem, the remainder of [tex]\( f(x) \)[/tex] when divided by [tex]\( (x-1) \)[/tex] can be found by evaluating [tex]\( f(1) \)[/tex].
[tex]\[
f(1) = 2(1)^2 + 5(1) + 6
\][/tex]
[tex]\[
f(1) = 2 \cdot 1 + 5 \cdot 1 + 6
\][/tex]
Simplify the expression:
[tex]\[
f(1) = 2 + 5 + 6
\][/tex]
[tex]\[
f(1) = 13
\][/tex]
Therefore, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( (x-1) \)[/tex] is [tex]\(\boxed{13}\)[/tex]. The correct option is (c) 13.
