1. **Problem:** Analyze the function $f(x) = \frac{x}{x-1}$. - The function is a rational function with a vertical asymptote at $x=1$ because the denominator is zero there. - The domain is all real numbers except $x=1$. 2. **Problem:** Analyze the function $f(x) = \frac{1}{x^2 - 1}$. - Factor the denominator: $x^2 - 1 = (x-1)(x+1)$. - Vertical asymptotes at $x=1$ and $x=-1$. - Domain excludes $x=\pm 1$. 3. **Problem:** Analyze the polynomial $f(x) = x^3 + 3x^2 + 2x$. - Factor: $f(x) = x(x^2 + 3x + 2) = x(x+1)(x+2)$. - Roots at $x=0, -1, -2$. - Domain is all real numbers. 4. **Problem:** Analyze $f(x) = \frac{1}{\sqrt{1 - x^2}}$. - The expression under the square root must be positive: $1 - x^2 > 0 \Rightarrow -1 < x < 1$. - Domain is $(-1,1)$. - Vertical asymptotes at $x=\pm 1$. 5. **Problem:** Analyze $f(x) = \sqrt{1 - x^2}$. - Domain: $1 - x^2 \geq 0 \Rightarrow -1 \leq x \leq 1$. - Range: $[0,1]$. 6. **Problem:** Analyze $f(x) = (x^2 - 1) \log x$. - Domain of $\log x$ is $x > 0$. - So domain is $x > 0$. - The factor $x^2 - 1$ is zero at $x=1$. 7. **Problem:** Analyze $f(x) = \frac{x}{e^{x-1}}$. - Domain is all real numbers. - Exponential in denominator never zero. 8. **Problem:** Analyze $f(x) = x^2 e^{-x}$. - Domain is all real numbers. - The function is a product of polynomial and exponential decay. **Final note:** Each function is studied for domain, roots, asymptotes, and behavior.