Question

the graph shows a logarithmic function f. what do f and the function g(x) = \log_{6}x have in common? select all that apply. \square both graphs increase on their domain. \square the graphs of both functions have the same x-intercept. \square neither graph has a y-intercept. \square neither graph has a vertical asymptote. \square both graphs have a horizontal asymptote. done

Explanation:

1. Both graphs increase on their domain: Logarithmic functions with a base \( b>1 \) (like \( g(x)=\log_6 x \), base \( 6 > 1 \)) are increasing. The graph of \( f \) also shows an increasing trend (as \( x \) increases, \( y \) increases). So this is true.
2. The graphs of both functions have the same x - intercept: For \( g(x)=\log_6 x \), the x - intercept is at \( x = 1 \) (since \( \log_6 1=0 \)). From the graph of \( f \), it also crosses the x - axis at \( x = 1 \), so they have the same x - intercept. This is true.
3. Neither graph has a y - intercept: The domain of a logarithmic function \( \log_b x \) is \( x>0 \). So \( x = 0 \) is not in the domain, meaning there is no \( y \) - intercept (since \( y \) - intercept occurs at \( x = 0 \)). For \( f \), the graph does not touch the \( y \) - axis (as \( x>0 \) for logarithmic functions), so neither has a \( y \) - intercept. This is true.
4. Neither graph has a vertical asymptote: Logarithmic functions \( \log_b x \) have a vertical asymptote at \( x = 0 \). The graph of \( f \) also approaches \( x = 0 \) (vertical line) as \( x \) approaches 0 from the right, so they do have a vertical asymptote. So this is false.
5. Both graphs have a horizontal asymptote: Logarithmic functions do not have horizontal asymptotes (they have vertical asymptotes). So this is false.

Answer:

• Both graphs increase on their domain.
• The graphs of both functions have the same x - intercept.
• Neither graph has a y - intercept.