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Given the definitions of f(x) and g(x) below, find the value of (f@g)(0). {:[f(x)=3x-3],[g(x)=3x^(2)-6x-1]:} Answer:

Given the definitions of f(x) f(x) and g(x) g(x) below, find the value of (f∘g)(0) (f \circ g)(0) .\newlinef(x)=3x−3g(x)=3x2−6x−1 \begin{array}{l} f(x)=3 x-3 \\ g(x)=3 x^{2}-6 x-1 \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 (f∘g)(0) (f \circ g)(0) .\newlinef(x)=3x−3g(x)=3x2−6x−1 \begin{array}{l} f(x)=3 x-3 \\ g(x)=3 x^{2}-6 x-1 \end{array} \newlineAnswer:
  1. Understand Notation: First, we need to understand the notation (f@g)(x)(f@g)(x). This notation means that we should first apply the function gg to xx, and then apply the function ff to the result of g(x)g(x). So, (f@g)(x)=f(g(x))(f@g)(x) = f(g(x)).
  2. Find g(0)g(0): Now, let's find g(0)g(0) by substituting x=0x = 0 into the function g(x)=3x2−6x−1g(x) = 3x^2 - 6x - 1.g(0)=3(0)2−6(0)−1=0−0−1=−1g(0) = 3(0)^2 - 6(0) - 1 = 0 - 0 - 1 = -1.
  3. Find f(g(0))f(g(0)): Next, we need to find f(g(0))f(g(0)) by substituting g(0)g(0) into the function f(x)=3x−3f(x) = 3x - 3. Since we found that g(0)=−1g(0) = -1, we substitute −1-1 for xx in f(x)f(x). f(g(0))=f(−1)=3(−1)−3=−3−3=−6f(g(0)) = f(-1) = 3(-1) - 3 = -3 - 3 = -6.
  4. Calculate (f@g)(0)(f@g)(0): Therefore, the value of (f@g)(0)(f@g)(0) is −6-6.

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