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Given the definitions of f(x) and g(x) below, find the value of (g@f)(-2). {:[f(x)=x^(2)-x-11],[g(x)=-2x-4]:} Answer:

Given the definitions of f(x) f(x) and g(x) g(x) below, find the value of (g∘f)(−2) (g \circ f)(-2) .\newlinef(x)=x2−x−11g(x)=−2x−4 \begin{array}{l} f(x)=x^{2}-x-11 \\ g(x)=-2 x-4 \end{array} \newlineAnswer:

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Q. Given the definitions of f(x) f(x) and g(x) g(x) below, find the value of (g∘f)(−2) (g \circ f)(-2) .\newlinef(x)=x2−x−11g(x)=−2x−4 \begin{array}{l} f(x)=x^{2}-x-11 \\ g(x)=-2 x-4 \end{array} \newlineAnswer:
  1. Understand Composition of Functions: Understand the composition of functions.\newlineThe notation g@f)(x) meansthatwefirstapply$fg@f)(x)\ means that we first apply \$f to xx, and then apply gg to the result of f(x)f(x). This is also written as g(f(x))g(f(x)).
  2. Calculate f(−2)f(-2): Calculate f(−2)f(-2).\newlineSubstitute −2-2 into f(x)f(x) to get f(−2)f(-2).\newlinef(−2)=(−2)2−(−2)−11f(-2) = (-2)^2 - (-2) - 11\newlinef(−2)=4+2−11f(-2) = 4 + 2 - 11\newlinef(−2)=6−11f(-2) = 6 - 11\newlinef(−2)=−5f(-2) = -5
  3. Calculate g(f(−2))g(f(-2)): Calculate g(f(−2))g(f(-2)).\newlineNow that we have f(−2)f(-2), we substitute −5-5 into g(x)g(x) to get g(f(−2))g(f(-2)).\newlineg(f(−2))=g(−5)g(f(-2)) = g(-5)\newlineg(−5)=−2(−5)−4g(-5) = -2(-5) - 4\newlineg(−5)=10−4g(-5) = 10 - 4\newlineg(−5)=6g(-5) = 6

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