Question

1. (0 points) graph the function. identify all asymptotes. plot at least two points on each graph.

(a) (f(x) = -\ln(x))
asymptotes:

(b) (f(x) = 3^x - 2)
asymptotes:

Explanation:

Step1: Find asymptote for (a)

$$\lim_{x \to 0^+} -\ln(x) = \infty \implies \text{Vertical asymptote: } x = 0$$

Step2: Find points for (a)

$$f(1) = -\ln(1) = 0 \implies (1, 0)$$

$$f(e) = -\ln(e) = -1 \implies (e, -1) \approx (2.72, -1)$$

Step3: Find asymptote for (b)

$$\lim_{x \to -\infty} (3^x - 2) = -2 \implies \text{Horizontal asymptote: } y = -2$$

Step4: Find points for (b)

$$f(0) = 3^0 - 2 = -1 \implies (0, -1)$$

$$f(1) = 3^1 - 2 = 1 \implies (1, 1)$$

Answer:

(a) Asymptote: Vertical asymptote \(x = 0\). Two points on the graph: \((1, 0)\) and \((e, -1) \approx (2.72, -1)\).
(b) Asymptote: Horizontal asymptote \(y = -2\). Two points on the graph: \((0, -1)\) and \((1, 1)\).