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Express your answer as a polynomial in standard form.f, of, x, equals, 3, x, squared, plus, 6, x, plus, 14f(x)=3x 2 +6x+14g, of, x, equals, x, minus, 6g(x)=x−6Find: , f, of, g, of, xFind: f(g(x))

Question

Express your answer as a polynomial in standard form.f, of, x, equals, 3, x, squared, plus, 6, x, plus, 14f(x)=3x 2 +6x+14g, of, x, equals, x, minus, 6g(x)=x−6Find: , f, of, g, of, xFind: f(g(x))

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Solution

Para encontrar f(g(x)) f(g(x)) , primero necesitamos sustituir g(x) g(x) en f(x) f(x) .

Dado: f(x)=3x2+6x+14 f(x) = 3x^2 + 6x + 14 g(x)=x−6 g(x) = x - 6

Primero, sustituimos g(x) g(x) en f(x) f(x) : f(g(x))=f(x−6) f(g(x)) = f(x - 6)

Ahora, sustituimos x−6 x - 6 en cada x x de f(x) f(x) : f(x−6)=3(x−6)2+6(x−6)+14 f(x - 6) = 3(x - 6)^2 + 6(x - 6) + 14

Expandimos y simplificamos: (x−6)2=x2−12x+36 (x - 6)^2 = x^2 - 12x + 36 3(x−6)2=3(x2−12x+36)=3x2−36x+108 3(x - 6)^2 = 3(x^2 - 12x + 36) = 3x^2 - 36x + 108

Ahora, sustituimos esto en la expresión: f(x−6)=3x2−36x+108+6(x−6)+14 f(x - 6) = 3x^2 - 36x + 108 + 6(x - 6) + 14

Expandimos y simplificamos: 6(x−6)=6x−36 6(x - 6) = 6x - 36 f(x−6)=3x2−36x+108+6x−36+14 f(x - 6) = 3x^2 - 36x + 108 + 6x - 36 + 14

Combinamos términos semejantes: f(x−6)=3x2−30x+86 f(x - 6) = 3x^2 - 30x + 86

Por lo tanto, la respuesta es: f(g(x))=3x2−30x+86 f(g(x)) = 3x^2 - 30x + 86

This problem has been solved

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