The functions \( f \) and \( g \) are defined on a suitable domain by \( f(x) = x^2 - 1 \) and \( g(x) = x^2 + 2 \).

A) Find the expression for \( f(g(x)) \).

B) Factorize \( f(g(x)) \).

f ( x ) = x 2 − 1 g ( x ) = x 2 + 2 f ( g ( x )) = ( x 2 + 2 ) 2 − 1 = x 4 + 4 x 2 + 4 − 1 = x 4 + 4 x 2 + 3 x 4 + 4 x 2 + 3 = x 4 + x 2 + 3 x 2 + 3 = x 2 ( x 2 + 1 ) + 3 ( x 2 + 1 ) = ( x 2 + 3 ) ( x 2 + 1 )

Answered by konrad509 | 2024-06-10

To find f ( g ( x )) , we substitute g ( x ) into f and expand, resulting in x 4 + 4 x 2 + 3 . This expression can be factored as ( x 2 + 3 ) ( x 2 + 1 ) .
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Answered by konrad509 | 2025-01-21