Question

the graph of a logarithmic function is shown. which interval represents all values of x for which the function is increasing? (-3, ∞) (-4, ∞) (2, ∞) (0, ∞)

Explanation:

Step1: Identify the vertical asymptote

The graph of the logarithmic function has a vertical asymptote. From the graph, we can see that the vertical asymptote is at \( x = -3 \) (since the graph approaches this line but doesn't cross it, and to the left of \( x=-3 \) the function is not defined in the domain of increase here).

Step2: Determine the interval of increase

For a logarithmic function, if it's increasing, we look at the domain where the function is defined and increasing. The graph starts from the vertical asymptote \( x = -3 \) and goes to the right (towards \( \infty \)) while increasing (since as \( x \) increases, \( y \) also increases). So the interval where the function is increasing is \( (-3, \infty) \).

Answer:

\((-3, \infty)\)