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Questions: Algebra BusinessCalculus
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If \(\displaystyle f(x)=\frac{3 x}{- 4 x^{2} - 4 x - 6}\) and \(\displaystyle g(x)=- x^{2} - 2 x - 8\) find \(\displaystyle (f\circ g)(x)\).
Evaluating \(\displaystyle f(x)\) at \(\displaystyle g(x)\) gives \(\displaystyle (f\circ g)(x)=f(g(x))=\frac{3 \left(- x^{2} - 2 x - 8\right)}{4 x^{2} + 8 x - 4 \left(- x^{2} - 2 x - 8\right)^{2} + 26}\).
\begin{question}If $f(x)=\frac{3 x}{- 4 x^{2} - 4 x - 6}$ and $g(x)=- x^{2} - 2 x - 8$ find $(f\circ g)(x)$.
\soln{9cm}{Evaluating $f(x)$ at $g(x)$ gives $(f\circ g)(x)=f(g(x))=\frac{3 \left(- x^{2} - 2 x - 8\right)}{4 x^{2} + 8 x - 4 \left(- x^{2} - 2 x - 8\right)^{2} + 26}$. }
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>If <img class="equation_image" title=" \displaystyle f(x)=\frac{3 x}{- 4 x^{2} - 4 x - 6} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D%5Cfrac%7B3%20x%7D%7B-%204%20x%5E%7B2%7D%20-%204%20x%20-%206%7D%20" alt="LaTeX: \displaystyle f(x)=\frac{3 x}{- 4 x^{2} - 4 x - 6} " data-equation-content=" \displaystyle f(x)=\frac{3 x}{- 4 x^{2} - 4 x - 6} " /> and <img class="equation_image" title=" \displaystyle g(x)=- x^{2} - 2 x - 8 " src="/equation_images/%20%5Cdisplaystyle%20g%28x%29%3D-%20x%5E%7B2%7D%20-%202%20x%20-%208%20" alt="LaTeX: \displaystyle g(x)=- x^{2} - 2 x - 8 " data-equation-content=" \displaystyle g(x)=- x^{2} - 2 x - 8 " /> find <img class="equation_image" title=" \displaystyle (f\circ g)(x) " src="/equation_images/%20%5Cdisplaystyle%20%28f%5Ccirc%20g%29%28x%29%20" alt="LaTeX: \displaystyle (f\circ g)(x) " data-equation-content=" \displaystyle (f\circ g)(x) " /> . </p> </p><p> <p>Evaluating <img class="equation_image" title=" \displaystyle f(x) " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20" alt="LaTeX: \displaystyle f(x) " data-equation-content=" \displaystyle f(x) " /> at <img class="equation_image" title=" \displaystyle g(x) " src="/equation_images/%20%5Cdisplaystyle%20g%28x%29%20" alt="LaTeX: \displaystyle g(x) " data-equation-content=" \displaystyle g(x) " /> gives <img class="equation_image" title=" \displaystyle (f\circ g)(x)=f(g(x))=\frac{3 \left(- x^{2} - 2 x - 8\right)}{4 x^{2} + 8 x - 4 \left(- x^{2} - 2 x - 8\right)^{2} + 26} " src="/equation_images/%20%5Cdisplaystyle%20%28f%5Ccirc%20g%29%28x%29%3Df%28g%28x%29%29%3D%5Cfrac%7B3%20%5Cleft%28-%20x%5E%7B2%7D%20-%202%20x%20-%208%5Cright%29%7D%7B4%20x%5E%7B2%7D%20%2B%208%20x%20-%204%20%5Cleft%28-%20x%5E%7B2%7D%20-%202%20x%20-%208%5Cright%29%5E%7B2%7D%20%2B%2026%7D%20" alt="LaTeX: \displaystyle (f\circ g)(x)=f(g(x))=\frac{3 \left(- x^{2} - 2 x - 8\right)}{4 x^{2} + 8 x - 4 \left(- x^{2} - 2 x - 8\right)^{2} + 26} " data-equation-content=" \displaystyle (f\circ g)(x)=f(g(x))=\frac{3 \left(- x^{2} - 2 x - 8\right)}{4 x^{2} + 8 x - 4 \left(- x^{2} - 2 x - 8\right)^{2} + 26} " /> . </p> </p>