10.) \( y=\ln (x-1) \)

To analyze the function \( y = \ln(x - 1) \), we can explore its features, such as its domain, range, intercepts, and asymptotes. ### Step 1: Determine the Domain The natural logarithm function \( \ln(x) \) is defined only for positive values. Therefore, for \( y = \ln(x - 1) \), we need \( x - 1 > 0 \). \[ x - 1 > 0 \implies x > 1 \] Thus, the domain of the function is \( (1, \infty) \). ### Step 2: Determine the Range The range of the natural logarithm function is all real numbers. Therefore, since \( y = \ln(x - 1) \) can take any real value as \( x \) approaches \( 1 \) from the right and goes to infinity, the range is: \[ (-\infty, \infty) \] ### Step 3: Find the Intercept To find the y-intercept, we set \( x = 0 \): \[ y = \ln(0 - 1) \quad \text{(undefined)} \] Since the function is undefined for \( x \leq 1 \), there is no y-intercept. ### Step 4: Find the Asymptote As \( x \) approaches \( 1 \) from the right, \( y \) approaches \( -\infty \). Therefore, there is a vertical asymptote at: \[ x = 1 \] ### Summary of Features - **Domain**: \( (1, \infty) \) - **Range**: \( (-\infty, \infty) \) - **Y-intercept**: None - **Vertical Asymptote**: \( x = 1 \) If you need further analysis or specific calculations, please let me know!