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What is (f∗g)(x)(f * g)(x)?\newlinef(x)=−x+4f(x) = -x + 4\newlineg(x)=4xg(x) = 4x\newline\newlineWrite your answer as a polynomial or a rational function in simplest form.\newline‾\underline{\hspace{3em}}\newline

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Full solution

Q. What is (f∗g)(x)(f * g)(x)?\newlinef(x)=−x+4f(x) = -x + 4\newlineg(x)=4xg(x) = 4x\newline\newlineWrite your answer as a polynomial or a rational function in simplest form.\newline‾\underline{\hspace{3em}}\newline
  1. Define functions f(x)f(x) and g(x)g(x): f(x)=−x+4f(x) = -x + 4 and g(x)=4xg(x) = 4x, so (f∗g)(x)=f(x)∗g(x)(f * g)(x) = f(x) * g(x).
  2. Expand (f∗g)(x)(f * g)(x): (f∗g)(x)=(–x+4)∗4x(f * g)(x) = (\text{–}x + 4) * 4x.
  3. Combine like terms: (f⋅g)(x)=−x⋅4x+4⋅4x(f \cdot g)(x) = -x \cdot 4x + 4 \cdot 4x.
  4. Final result: (f∗g)(x)=−4x2+16x(f * g)(x) = -4x^2 + 16x.

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