Question

given \\(f(x) = 2x^2 + 3\\) and \\(g(x) = 4x - 5\\), the composition \\(f(g(x))\\) is equal to \\(32x^2 - 80x + 53\\).

now find an expression for \\(g(f(x))\\).

\\(g(f(x)) = \square x^2 + \square\\)

Explanation:

🆕 New Concept Discovered: Composition of Functions
Plugging one function inside another.

Step 1: Identify the outer and inner functions

To find \( g(f(x)) \), we use \( g(x) \) as the outer function and substitute the entire expression for \( f(x) \) wherever there is an \( x \) in \( g(x) \).

Given:

$$ f(x) = 2x^2 + 3 $$

$$ g(x) = 4x - 5 $$

Step 2: Substitute \( f(x) \) into \( g(x) \)

Replace \( x \) in \( g(x) \) with the expression for \( f(x) \):

$$ g(f(x)) = 4(2x^2 + 3) - 5 $$

Step 3: Simplify the expression

Distribute the \( 4 \) and combine like terms:

$$ g(f(x)) = 8x^2 + 12 - 5 $$

$$ g(f(x)) = 8x^2 + 7 $$

Answer:

$$ g(f(x)) = 8x^2 + 7 $$