Find [tex]$f(k-1)$[/tex] when [tex]$f(x) = 3x^2 + 4x + 5$[/tex].

Evaluate:
A. [tex][tex]$3(k-1)^2 + 4(k-1) + 5$[/tex][/tex]
B. All of the above
C. [tex]$3k^2 - 2k - 4$[/tex]
D. None of the above



Answer :

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]