Question

consider the two functions.
f(x) = x² - 4x + 1
g(x) = 2x - 3
which expression represents g(f(x))?
○ 2x² - 4x - 2
○ 2x² - 8x - 1
○ 4x² - 8x - 5
○ 4x² - 20x + 22

Explanation:

Step1: Understand Function Composition

To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \). Given \( f(x) = x^2 - 4x + 1 \) and \( g(x) = 2x - 3 \), we replace \( x \) in \( g(x) \) with \( f(x) \).

Step2: Substitute \( f(x) \) into \( g(x) \)

Substitute \( x = x^2 - 4x + 1 \) into \( g(x) = 2x - 3 \):

$$ g(f(x)) = 2(x^2 - 4x + 1) - 3 $$

Step3: Distribute and Simplify

First, distribute the 2:

$$ 2(x^2 - 4x + 1) = 2x^2 - 8x + 2 $$

Then subtract 3:

$$ 2x^2 - 8x + 2 - 3 = 2x^2 - 8x - 1 $$

Answer:

\( 2x^2 - 8x - 1 \) (corresponding to the option "2x² - 8x - 1")