Question

select all the true statements about the function y = -ln x. the function has a y - intercept at (0, 1). the function has an x - intercept at (1, 0). the function has a vertical asymptote at x = 0. the function has a horizontal asymptote at y = 0.

Explanation:

Step1: Check x - intercept

Set y = 0, so 0=-ln x. Then ln x = 0. Since ln 1 = 0, x = 1. So the x - intercept is (1,0).

Step2: Check vertical asymptote

The domain of y = -ln x is x>0. As x approaches 0 from the right, -ln x approaches +∞. So x = 0 is a vertical asymptote.

Step3: Check horizontal asymptote

As x→+∞, -ln x→ -∞. But if we consider the behavior in terms of limits for horizontal asymptotes, we know that \(\lim_{x
ightarrow+\infty}- \ln x=-\infty\) and \(\lim_{x
ightarrow0^{+}}-\ln x = +\infty\). However, if we rewrite the function and consider the general form of the natural - log function's behavior, we note that as \(x
ightarrow+\infty\), the function \(y =-\ln x\) gets closer and closer to \(y = 0\) from below. So \(y = 0\) is a horizontal asymptote.

Step4: Check y - intercept

The function \(y=-\ln x\) is not defined for \(x = 0\), so there is no y - intercept at \((0,1)\) or any other point with \(x = 0\).

Answer:

1. The function has an x - intercept at (1, 0).
2. The function has a vertical asymptote at x = 0.
3. The function has a horizontal asymptote at y = 0.