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{:[g(x)=3-x^(2)],[h(x)=4-3x]:} Write (g@h)(x) as an expression in terms of x. (g@h)(x)=

g(x)=3−x2h(x)=4−3x \begin{array}{l} g(x)=3-x^{2} \\ h(x)=4-3 x \end{array} \newlineWrite (g∘h)(x) (g \circ h)(x) as an expression in terms of x x .\newline(g∘h)(x)= (g \circ h)(x)=

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Q. g(x)=3−x2h(x)=4−3x \begin{array}{l} g(x)=3-x^{2} \\ h(x)=4-3 x \end{array} \newlineWrite (g∘h)(x) (g \circ h)(x) as an expression in terms of x x .\newline(g∘h)(x)= (g \circ h)(x)=
  1. Find composition of functions: We need to find the composition of the functions g(x)g(x) and h(x)h(x), which is written as (g@h)(x)(g@h)(x). This means we need to substitute the function h(x)h(x) into the function g(x)g(x).
  2. Write down g(x)g(x) and h(x)h(x): First, let's write down the functions g(x)g(x) and h(x)h(x) for clarity:\newlineg(x)=3−x2g(x) = 3 - x^2\newlineh(x)=4−3xh(x) = 4 - 3x\newlineNow, to find (g@h)(x)(g@h)(x), we replace every instance of xx in g(x)g(x) with h(x)h(x).
  3. Perform substitution: Perform the substitution:\newline(g@h)(x)=g(h(x))=3−(h(x))2(g@h)(x) = g(h(x)) = 3 - (h(x))^2\newlineNow, we need to square the function h(x)h(x).
  4. Square h(x)h(x): Square h(x)h(x):(h(x))2=(4−3x)2(h(x))^2 = (4 - 3x)^2To square a binomial, we use the formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
  5. Apply formula to square h(x)h(x): Apply the formula to square h(x)h(x):(4−3x)2=42−2×(4)×(3x)+(3x)2(4 - 3x)^2 = 4^2 - 2\times(4)\times(3x) + (3x)^2Calculate the square and products:(4−3x)2=16−24x+9x2(4 - 3x)^2 = 16 - 24x + 9x^2
  6. Calculate square and products: Now, substitute (4−3x)2(4 - 3x)^2 back into the expression for (g@h)(x)(g@h)(x):\newline(g@h)(x)=3−(16−24x+9x2)(g@h)(x) = 3 - (16 - 24x + 9x^2)\newlineSimplify the expression by distributing the negative sign:\newline(g@h)(x)=3−16+24x−9x2(g@h)(x) = 3 - 16 + 24x - 9x^2
  7. Substitute back into (g@h)(x)(g@h)(x): Combine like terms:\newline(g@h)(x)=−13+24x−9x2(g@h)(x) = -13 + 24x - 9x^2\newlineReorder the terms to write the quadratic in standard form:\newline(g@h)(x)=−9x2+24x−13(g@h)(x) = -9x^2 + 24x - 13

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