Which of the following are key attributes of y=Inx. I. The graph has a vertical asymptote at x=0 II. The graph has a horizontal asymptote at y=2.5 III. The graph goes through the point (e,0) IV. The graph has a domain of [0,infinity]
Solution
Step 1 ：The question is asking about the properties of the natural logarithm function, y=ln(x).
Step 2 ：The graph of y=ln(x) does have a vertical asymptote at x=0. This is because as x approaches 0 from the right, ln(x) approaches negative infinity.
Step 3 ：The graph of y=ln(x) does not have a horizontal asymptote at y=2.5. In fact, it does not have any horizontal asymptotes. As x approaches infinity, ln(x) also approaches infinity.
Step 4 ：The graph of y=ln(x) does go through the point (e,1), not (e,0). This is because ln(e) = 1.
Step 5 ：The graph of y=ln(x) does have a domain of (0,infinity), not [0,infinity]. This is because ln(x) is undefined for x<=0.
Step 6 ：The key attributes of y=ln(x) are I and IV. However, for IV, the domain should be (0,infinity), not [0,infinity]. So, the correct answer is \(\boxed{\text{I and IV with correction}}\).
