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Express your answer as a polynomial in standard form.f, of, x, equals, 2, x, squared, plus, 3, x, plus, 12f(x)=2x 2 +3x+12g, of, x, equals, 5, x, minus, 6g(x)=5x−6Find: , left parenthesis, g, circle, f, right parenthesis, left parenthesis, x, right parenthesisFind: (g∘f)(x)

Question

Express your answer as a polynomial in standard form.f, of, x, equals, 2, x, squared, plus, 3, x, plus, 12f(x)=2x 2 +3x+12g, of, x, equals, 5, x, minus, 6g(x)=5x−6Find: , left parenthesis, g, circle, f, right parenthesis, left parenthesis, x, right parenthesisFind: (g∘f)(x)

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Solution

Para encontrar (gf)(x)(g \circ f)(x), primero necesitamos entender que esto significa g(f(x))g(f(x)). Esto implica que debemos sustituir f(x)f(x) en la función g(x)g(x).

Dado: f(x)=2x2+3x+12 f(x) = 2x^2 + 3x + 12 g(x)=5x6 g(x) = 5x - 6

Primero, evaluamos gg en f(x)f(x): g(f(x))=g(2x2+3x+12) g(f(x)) = g(2x^2 + 3x + 12)

Ahora sustituimos 2x2+3x+122x^2 + 3x + 12 en g(x)g(x): g(2x2+3x+12)=5(2x2+3x+12)6 g(2x^2 + 3x + 12) = 5(2x^2 + 3x + 12) - 6

Distribuimos el 5: =52x2+53x+5126 = 5 \cdot 2x^2 + 5 \cdot 3x + 5 \cdot 12 - 6 =10x2+15x+606 = 10x^2 + 15x + 60 - 6

Simplificamos: =10x2+15x+54 = 10x^2 + 15x + 54

Por lo tanto, (gf)(x)=10x2+15x+54(g \circ f)(x) = 10x^2 + 15x + 54.

This problem has been solved

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