Questions: Which function has an asymptote at x=5 and an x-intercept of (6,0)? A. f(x)=log (x-5) B. f(x)=log (x+5) C. f(x)=log x-5 D. f(x)=log x+5

Solution Steps

To determine which function has an asymptote at \( x = 5 \) and an \( x \)-intercept at \( (6,0) \), we need to analyze the properties of logarithmic functions. The vertical asymptote of a logarithmic function \( f(x) = \log(x - a) \) occurs at \( x = a \). The \( x \)-intercept occurs where \( f(x) = 0 \), which means solving \( \log(x - a) = 0 \).

1. Identify the vertical asymptote for each function.
2. Check which function has an asymptote at \( x = 5 \).
3. Verify the \( x \)-intercept by solving \( \log(x - a) = 0 \) for \( x \).

For a logarithmic function of the form \( f(x) = \log(x - a) \), the vertical asymptote occurs at \( x = a \). We need to find a function with an asymptote at \( x = 5 \). This corresponds to the function \( f(x) = \log(x - 5) \).

The \( x \)-intercept occurs where \( f(x) = 0 \). For the function \( f(x) = \log(x - 5) \), we set up the equation: \[ \log(x - 5) = 0 \] This implies: \[ x - 5 = 1 \quad \Rightarrow \quad x = 6 \] Thus, the \( x \)-intercept is at the point \( (6, 0) \).

We check the other options to confirm they do not meet the criteria:

• Option B: \( f(x) = \log(x + 5) \) has no asymptote at \( x = 5 \).
• Option C: \( f(x) = \log(x) - 5 \) has no asymptote at \( x = 5 \).
• Option D: \( f(x) = \log(x) + 5 \) has no asymptote at \( x = 5 \).

Final Answer