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.

both graphs increase on their domain.
the graphs of both functions have the same x-intercept.
neither graph has a y-intercept.
neither graph has a vertical asymptote.
both graphs have a horizontal asymptote.

Explanation:

Analyze the graph of function f

Using the Logarithmic Functions knowledge point

• The graph of \(f\) passes through \((1, 0)\) and increases from left to right.
• The domain of \(f\) is \((0, \infty)\), and it has a vertical asymptote at \(x = 0\) (the y-axis).
• There is no y-intercept, and there is no horizontal asymptote.

Analyze the function g(x)

Using the Logarithmic Functions knowledge point

• The function \(g(x) = \log_6 x\) is a standard logarithmic function with base \(6 > 1\).
• Its domain is \((0, \infty)\), and it increases on its entire domain.
• Its x-intercept is at \((1, 0)\) since \(\log_6(1) = 0\).
• It has a vertical asymptote at \(x = 0\), meaning it has no y-intercept.
• It has no horizontal asymptote.

Evaluate the given statements

Using the Logarithmic Intercepts and Asymptotic Behavior of Logarithms knowledge points

• Both graphs increase on their domain: True. Both \(f\) and \(g\) are increasing logarithmic functions.
• The graphs of both functions have the same x-intercept: True. Both graphs cross the x-axis at \((1, 0)\).
• Neither graph has a y-intercept: True. Since the domain for both is \(x > 0\), neither graph crosses the y-axis.
• Neither graph has a vertical asymptote: False. Both have a vertical asymptote at \(x = 0\).
• Both graphs have a horizontal asymptote: False. Logarithmic functions grow without bound and do not have horizontal asymptotes.

Answer:

• Both graphs increase on their domain. (Correct answer)
• The graphs of both functions have the same x-intercept. (Correct answer)
• Neither graph has a y-intercept. (Correct answer)
• Neither graph has a vertical asymptote.
• Both graphs have a horizontal asymptote.