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Given the definitions of f(x) and g(x) below, find the value of (g@f)(-2). {:[f(x)=-x-5],[g(x)=2x^(2)+4x-6]:} 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)=−x−5g(x)=2x2+4x−6 \begin{array}{l} f(x)=-x-5 \\ g(x)=2 x^{2}+4 x-6 \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)=−x−5g(x)=2x2+4x−6 \begin{array}{l} f(x)=-x-5 \\ g(x)=2 x^{2}+4 x-6 \end{array} \newlineAnswer:
  1. Understand composition of functions: Understand the composition of functions. The composition of functions g∘f)(x) meanswefirstapply$fg\circ f)(x)\ means we first apply \$f to xx, and then apply gg to the result of f(x)f(x). In notation, this is g(f(x))g(f(x)).
  2. Calculate f(−2)f(-2): Calculate f(−2)f(-2). We have f(x)=−x−5f(x) = -x - 5. To find f(−2)f(-2), we substitute xx with −2-2. f(−2)=−(−2)−5=2−5=−3f(-2) = -(-2) - 5 = 2 - 5 = -3.
  3. Calculate g(f(−2))g(f(-2)): Calculate g(f(−2))g(f(-2)).\newlineNow that we have f(−2)=−3f(-2) = -3, we need to find g(−3)g(-3). The function g(x)g(x) is given by g(x)=2x2+4x−6g(x) = 2x^2 + 4x - 6.\newlineSubstitute xx with −3-3 into g(x)g(x).\newlineg(−3)=2(−3)2+4(−3)−6g(-3) = 2(-3)^2 + 4(-3) - 6.
  4. Simplify g(−3)g(-3): Simplify g(−3)g(-3).\newlineCalculate the value of g(−3)g(-3) step by step.\newlineg(−3)=2(9)+4(−3)−6g(-3) = 2(9) + 4(-3) - 6\newlineg(−3)=18−12−6g(-3) = 18 - 12 - 6\newlineg(−3)=6−6g(-3) = 6 - 6\newlineg(−3)=0g(-3) = 0.

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