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{:[f(x)=6-(1)/(2)x],[h(x)=4(x-3)^(2)]:} Write f(h(x)) as an expression in terms of x. f(h(x))=

f(x)=6−12xh(x)=4(x−3)2 \begin{array}{l} f(x)=6-\frac{1}{2} x \\ h(x)=4(x-3)^{2} \end{array} \newlineWrite f(h(x)) f(h(x)) as an expression in terms of x x .\newlinef(h(x))= f(h(x))=

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Q. f(x)=6−12xh(x)=4(x−3)2 \begin{array}{l} f(x)=6-\frac{1}{2} x \\ h(x)=4(x-3)^{2} \end{array} \newlineWrite f(h(x)) f(h(x)) as an expression in terms of x x .\newlinef(h(x))= f(h(x))=
  1. Identify functions and composition: Identify the given functions and the composition we need to find.\newlineWe have two functions:\newlinef(x)=6−12xf(x) = 6 - \frac{1}{2}x\newlineh(x)=4(x−3)2h(x) = 4(x - 3)^2\newlineWe need to find the composition f(h(x))f(h(x)), which means we will substitute h(x)h(x) into f(x)f(x) where xx is.
  2. Substitute h(x)h(x) into f(x)f(x): Substitute h(x)h(x) into f(x)f(x).\newlineTo find f(h(x))f(h(x)), we replace every xx in f(x)f(x) with h(x)h(x):\newlinef(h(x))=6−(12)h(x)f(h(x)) = 6 - \left(\frac{1}{2}\right)h(x)\newlineNow we need to substitute the expression for h(x)h(x) into this new function.
  3. Perform the substitution: Perform the substitution.\newlineSubstitute h(x)=4(x−3)2h(x) = 4(x - 3)^2 into the expression for f(h(x))f(h(x)):\newlinef(h(x))=6−12(4(x−3)2)f(h(x)) = 6 - \frac{1}{2}(4(x - 3)^2)
  4. Simplify the expression: Simplify the expression.\newlineNow we simplify the expression by distributing the 12\frac{1}{2} into 4(x−3)24(x - 3)^2:\newlinef(h(x))=6−2(x−3)2f(h(x)) = 6 - 2(x - 3)^2
  5. Expand the squared term: Expand the squared term (if necessary).\newlineIn this case, we can leave the expression in its factored form, as it is already simplified. So, the final expression for f(h(x))f(h(x)) is:\newlinef(h(x))=6−2(x−3)2f(h(x)) = 6 - 2(x - 3)^2

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