To find [tex]\( f(k-1) \)[/tex] when [tex]\( f(x) = 3x^2 + 4x + 5 \)[/tex], follow these steps:
1. Substitute [tex]\((k-1)\)[/tex] into the function [tex]\( f(x) \)[/tex]. This involves replacing [tex]\( x \)[/tex] with [tex]\( (k-1) \)[/tex] in the function's expression.
[tex]\[
f(k-1) = 3(k-1)^2 + 4(k-1) + 5
\][/tex]
2. Expand and simplify the expression.
First, expand [tex]\((k-1)^2\)[/tex]:
[tex]\[
(k-1)^2 = k^2 - 2k + 1
\][/tex]
Now substitute this back into the expression:
[tex]\[
f(k-1) = 3(k^2 - 2k + 1) + 4(k-1) + 5
\][/tex]
3. Distribute the constants and combine like terms.
[tex]\[
f(k-1) = 3k^2 - 6k + 3 + 4k - 4 + 5
\][/tex]
4. Combine all the terms:
[tex]\[
3k^2 - 6k + 3 + 4k - 4 + 5 = 3k^2 + (-6k + 4k) + (3 - 4 + 5)
\][/tex]
Simplify the coefficients:
[tex]\[
f(k-1) = 3k^2 - 2k + 4
\][/tex]
Hence, the correct answer is:
(A) [tex]\( 3k^2 - 2k + 4 \)[/tex]
