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Express your answer as a polynomial in standard form.f, of, x, equals, x, squared, plus, 6, x, plus, 8f(x)=x 2 +6x+8g, of, x, equals, minus, 3, x, minus, 10g(x)=−3x−10Find: , left parenthesis, f, circle, g, right parenthesis, left parenthesis, x, right parenthesisFind: (f∘g)(x)

Question

Express your answer as a polynomial in standard form.f, of, x, equals, x, squared, plus, 6, x, plus, 8f(x)=x 2 +6x+8g, of, x, equals, minus, 3, x, minus, 10g(x)=−3x−10Find: , left parenthesis, f, circle, g, right parenthesis, left parenthesis, x, right parenthesisFind: (f∘g)(x)

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Solution

Para encontrar (f∘g)(x)(f \circ g)(x), primero debemos entender que esto significa f(g(x))f(g(x)). Es decir, debemos sustituir g(x)g(x) en la función f(x)f(x).

Dado: f(x)=x2+6x+8f(x) = x^2 + 6x + 8 g(x)=−3x−10g(x) = -3x - 10

Primero, sustituimos g(x)g(x) en f(x)f(x):

f(g(x))=f(−3x−10)f(g(x)) = f(-3x - 10)

Ahora, sustituimos −3x−10-3x - 10 en cada xx de f(x)f(x):

f(−3x−10)=(−3x−10)2+6(−3x−10)+8f(-3x - 10) = (-3x - 10)^2 + 6(-3x - 10) + 8

Desarrollamos cada término:

  1. (−3x−10)2(-3x - 10)^2: (−3x−10)2=(−3x)2+2(−3x)(−10)+(−10)2=9x2+60x+100 (-3x - 10)^2 = (-3x)^2 + 2(-3x)(-10) + (-10)^2 = 9x^2 + 60x + 100

  2. 6(−3x−10)6(-3x - 10): 6(−3x−10)=−18x−60 6(-3x - 10) = -18x - 60

Ahora sumamos todos los términos:

f(−3x−10)=9x2+60x+100−18x−60+8 f(-3x - 10) = 9x^2 + 60x + 100 - 18x - 60 + 8

Simplificamos combinando términos semejantes:

f(−3x−10)=9x2+(60x−18x)+(100−60+8) f(-3x - 10) = 9x^2 + (60x - 18x) + (100 - 60 + 8) f(−3x−10)=9x2+42x+48 f(-3x - 10) = 9x^2 + 42x + 48

Por lo tanto, (f∘g)(x)=9x2+42x+48(f \circ g)(x) = 9x^2 + 42x + 48.

This problem has been solved

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