9Number 9ExplanationGiven functions f:R→Rf: \mathbb{R} \rightarrow \mathbb{R}f:R→R and g:R→Rg: \mathbb{R} \rightarrow \mathbb{R}g:R→R defined by f(x)=2x2−3f(x) = 2x^2 - 3f(x)=2x2−3 and g(x)=3x−1g(x) = 3x - 1g(x)=3x−1 The composite function (f∘g)(x)(f \circ g)(x)(f∘g)(x) is defined as ....(f∘g)(x)=18x2−12x−21(f \circ g)(x) = 18x^2 - 12x - 21(f∘g)(x)=18x2−12x−21(f∘g)(x)=18x2+10x−21(f \circ g)(x) = 18x^2 + 10x - 21(f∘g)(x)=18x2+10x−21(f∘g)(x)=18x2−12x−1(f \circ g)(x) = 18x^2 - 12x - 1(f∘g)(x)=18x2−12x−1(f∘g)(x)=9x2−6x−2(f \circ g)(x) = 9x^2 - 6x - 2(f∘g)(x)=9x2−6x−2(f∘g)(x)=9x2−6x+2(f \circ g)(x) = 9x^2 - 6x + 2(f∘g)(x)=9x2−6x+2ExplanationGiven f(x)=2x2−3f(x) = 2x^2 - 3f(x)=2x2−3 and g(x)=3x−1g(x) = 3x - 1g(x)=3x−1 The composite function (f∘g)(x)(f \circ g)(x)(f∘g)(x) is f(g(x))f(g(x))f(g(x)) Substitute function g(x)g(x)g(x) into function f(x)f(x)f(x) (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))(f∘g)(x)=f(g(x))=2(g(x))2−3= 2(g(x))^2 - 3=2(g(x))2−3=2(3x−1)2−3= 2(3x - 1)^2 - 3=2(3x−1)2−3 Expand (3x−1)2(3x - 1)^2(3x−1)2 using the formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2(a−b)2=a2−2ab+b2 =2(9x2−6x+1)−3= 2(9x^2 - 6x + 1) - 3=2(9x2−6x+1)−3=18x2−12x+2−3= 18x^2 - 12x + 2 - 3=18x2−12x+2−3=18x2−12x−1= 18x^2 - 12x - 1=18x2−12x−1 Therefore, the composite function (f∘g)(x)=18x2−12x−1(f \circ g)(x) = 18x^2 - 12x - 1(f∘g)(x)=18x2−12x−1
