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f(x)=2x+3 g(x)=x^(2)-3x+1 Write g(f(x)) as an expression in terms of x. g(f(x))=

f(x)=2x+3 f(x)=2 x+3 \newlineg(x)=x2−3x+1 g(x)=x^{2}-3 x+1 \newlineWrite g(f(x)) g(f(x)) as an expression in terms of x x .\newlineg(f(x))= g(f(x))=

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Q. f(x)=2x+3 f(x)=2 x+3 \newlineg(x)=x2−3x+1 g(x)=x^{2}-3 x+1 \newlineWrite g(f(x)) g(f(x)) as an expression in terms of x x .\newlineg(f(x))= g(f(x))=
  1. Identify functions: Identify the functions to be composed.\newlineWe have two functions:\newlinef(x)=2x+3f(x) = 2x + 3\newlineg(x)=x2−3x+1g(x) = x^2 - 3x + 1\newlineWe need to find the composition of g(f(x))g(f(x)), which means we will substitute f(x)f(x) into g(x)g(x).
  2. Substitute f(x)f(x) into g(x)g(x): Substitute f(x)f(x) into g(x)g(x). To find g(f(x))g(f(x)), we replace every instance of xx in g(x)g(x) with the expression for f(x)f(x). So, g(f(x))=(2x+3)2−3(2x+3)+1g(f(x)) = (2x + 3)^2 - 3(2x + 3) + 1
  3. Expand (2x+3)2(2x + 3)^2: Expand the expression (2x+3)2.(2x + 3)^2.(2x+3)2=(2x+3)(2x+3)=4x2+6x+6x+9=4x2+12x+9(2x + 3)^2 = (2x + 3)(2x + 3) = 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9
  4. Distribute −3-3: Distribute the −3-3 across the terms in the parentheses.\newline−3(2x+3)=−6x−9-3(2x + 3) = -6x - 9
  5. Combine terms: Combine all the terms to get the final expression for g(f(x))g(f(x)).g(f(x))=4x2+12x+9−6x−9+1g(f(x)) = 4x^2 + 12x + 9 - 6x - 9 + 1
  6. Simplify expression: Simplify the expression by combining like terms.\newlineg(f(x))=4x2+(12x−6x)+(9−9+1)g(f(x)) = 4x^2 + (12x - 6x) + (9 - 9 + 1)\newlineg(f(x))=4x2+6x+1g(f(x)) = 4x^2 + 6x + 1

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