Question

1 mark for review the functions f and g are given by f(x) = x² + 1 and g(x) = 4x - 1. which of the following is an expression for f(g(x))? a x² + 4x b 4x² + 3 c 16x² - 8x + 2 d 4x³ - x² + 4x - 1

Explanation:

Step1: Understand function composition

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \). That is, replace every \( x \) in \( f(x) \) with \( g(x) \).

Given \( f(x) = x^2 + 1 \) and \( g(x) = 4x - 1 \), so we substitute \( g(x) = 4x - 1 \) into \( f \).

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

\( f(g(x)) = f(4x - 1) \)

Now, replace \( x \) in \( f(x) = x^2 + 1 \) with \( 4x - 1 \):

\( f(4x - 1) = (4x - 1)^2 + 1 \)

Step3: Expand \( (4x - 1)^2 \)

Using the formula \( (a - b)^2 = a^2 - 2ab + b^2 \), where \( a = 4x \) and \( b = 1 \):

\( (4x - 1)^2 = (4x)^2 - 2(4x)(1) + 1^2 = 16x^2 - 8x + 1 \)

Step4: Add 1 to the expanded form

\( f(4x - 1) = 16x^2 - 8x + 1 + 1 = 16x^2 - 8x + 2 \)

Answer:

C. \( 16x^2 - 8x + 2 \)