Jika diketahui f(x) = 4x−1 dan g(x) = 2x2−3x+4 , maka fungsi gof(x) = .... 32x2-x+5 9x2-32x+28 32x2-28x+9 28x2-32x+9 28x2-32x-9

Mathematics

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Answer

C. 32x2−28x+9\text{C. } 32x^2-28x+9

Step-by-step solution

4 steps
  1. Understand the definition of the composite function g∘f(x)g \circ f(x).

    The notation g∘f(x)g \circ f(x) means g(f(x))g(f(x)), which implies substituting the entire function f(x)f(x) into the variable xx of the function g(x)g(x).

  2. Substitute f(x)f(x) into g(x)g(x).

    Given:
    f(x)=4x−1f(x) = 4x - 1
    g(x)=2x2−3x+4g(x) = 2x^2 - 3x + 4

    Replace every xx in g(x)g(x) with f(x)f(x):

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

    Now substitute f(x)=4x−1f(x) = 4x - 1:

    g(f(x))=2(4x−1)2−3(4x−1)+4g(f(x)) = 2(4x - 1)^2 - 3(4x - 1) + 4
  3. Expand and simplify the expression.

    First, expand the squared term (4x−1)2(4x - 1)^2:

    (4x−1)2=(4x)2−2(4x)(1)+(1)2=16x2−8x+1(4x - 1)^2 = (4x)^2 - 2(4x)(1) + (1)^2 = 16x^2 - 8x + 1

    Now substitute this back into the expression:

    g(f(x))=2(16x2−8x+1)−3(4x−1)+4g(f(x)) = 2(16x^2 - 8x + 1) - 3(4x - 1) + 4

    Distribute the constants:

    g(f(x))=(32x2−16x+2)−(12x−3)+4g(f(x)) = (32x^2 - 16x + 2) - (12x - 3) + 4 g(f(x))=32x2−16x+2−12x+3+4g(f(x)) = 32x^2 - 16x + 2 - 12x + 3 + 4

    Combine like terms:

    g(f(x))=32x2+(−16x−12x)+(2+3+4)g(f(x)) = 32x^2 + (-16x - 12x) + (2 + 3 + 4) g(f(x))=32x2−28x+9g(f(x)) = 32x^2 - 28x + 9
  4. Compare the result with the given options.

    The calculated function is 32x2−28x+932x^2 - 28x + 9. This matches option C.

    The final answer is C. 32x2−28x+9\text{C. } 32x^2-28x+9.

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