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

g(x)=x2+6xh(x)=x2−1 \begin{array}{l} g(x)=x^{2}+6 x \\ h(x)=x^{2}-1 \end{array} \newlineWrite g(h(x)) g(h(x)) as an expression in terms of x x .\newlineg(h(x))= g(h(x))=

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Q. g(x)=x2+6xh(x)=x2−1 \begin{array}{l} g(x)=x^{2}+6 x \\ h(x)=x^{2}-1 \end{array} \newlineWrite g(h(x)) g(h(x)) as an expression in terms of x x .\newlineg(h(x))= g(h(x))=
  1. Identify Functions and Task: Identify the functions given and what is being asked.\newlineWe have two functions:\newlineg(x)=x2+6xg(x) = x^2 + 6x\newlineh(x)=x2−1h(x) = x^2 - 1\newlineWe need to find the composition of these functions, which is g(h(x))g(h(x)).
  2. Substitute h(x)h(x) into g(x)g(x): Substitute h(x)h(x) into g(x)g(x).\newlineTo find g(h(x))g(h(x)), we replace every instance of xx in g(x)g(x) with h(x)h(x):\newlineg(h(x))=(h(x))2+6⋅h(x)g(h(x)) = (h(x))^2 + 6 \cdot h(x)
  3. Plug in h(x)h(x) into Equation: Plug in the expression for h(x)h(x) into the equation from Step 22.\newlineg(h(x))=(x2−1)2+6⋅(x2−1)g(h(x)) = (x^2 - 1)^2 + 6 \cdot (x^2 - 1)
  4. Expand and Distribute: Expand the squared term and distribute the 66.g(h(x))=(x2−1)(x2−1)+6x2−6g(h(x)) = (x^2 - 1)(x^2 - 1) + 6x^2 - 6g(h(x))=x4−2x2+1+6x2−6g(h(x)) = x^4 - 2x^2 + 1 + 6x^2 - 6
  5. Combine Like Terms: Combine like terms. g(h(x))=x4+4x2−5g(h(x)) = x^4 + 4x^2 - 5

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