For the following function, find the intervals on which the function is increasing or decreasing. \( f(x)=\frac{x+9}{x+1} \)

Analyze the function features with the analysis type as "intervals" for the function \( f(x)=\frac{x+9}{x+1} \). Function by following steps: - step0: Find the increasing or decreasing interval: \(f\left(x\right)=\frac{x+9}{x+1}\) - step1: Find the domain: \(f\left(x\right)=\frac{x+9}{x+1},x\neq -1\) - step2: Find the derivative: \(f^{\prime}\left(x\right)=-\frac{8}{\left(x+1\right)^{2}}\) - step3: Find the domain: \(f^{\prime}\left(x\right)=-\frac{8}{\left(x+1\right)^{2}},x\neq -1\) - step4: Substitute \(f^{\prime}\left(x\right)=0:\) \(0=-\frac{8}{\left(x+1\right)^{2}}\) - step5: Swap the sides: \(-\frac{8}{\left(x+1\right)^{2}}=0\) - step6: Rewrite the expression: \(\frac{-8}{\left(x+1\right)^{2}}=0\) - step7: Cross multiply: \(-8=\left(x+1\right)^{2}\times 0\) - step8: Simplify the equation: \(-8=0\) - step9: The statement is false: \(x \in \varnothing \) - step10: Determine the intervals: \(x \in \left(-\infty,-1\right)\cup \left(-1,+\infty\right)\) - step11: Choose the points: \(x_{1}=-2\) - step12: Find the values of the derivatives: \(f^{\prime}\left(-2\right)=-8\) - step13: Calculate: \(\begin{align}&x \in \left(-\infty,-1\right)\cup \left(-1,+\infty\right)\textrm{ is decreasing interval}\end{align}\) - step14: Evaluate: \(\begin{align}&\textrm{No increasing interval}\\&\textrm{The decreasing interval is}\textrm{ }x \in \left(-\infty,-1\right)\cup \left(-1,+\infty\right)\end{align}\) The function \( f(x)=\frac{x+9}{x+1} \) is decreasing on the interval \( x \in (-\infty,-1) \cup (-1,+\infty) \).