Question

1. which was more difficult, exam 1 or exam 2? why?
2. what did you do better in preparing for exam 2 than for exam 1?
3. exponential growth and decay.

a. have you ever heard of exponential growth?
b. what about exponential decay?
c. what do those phrases mean to you?
d. can you think of an example of something that grows exponentially?
e. can you think of an example of something that decays exponentially?

1. let (f(x) = 7 - e^x) and (g(x) = \ln(7 - x)).

a. what are the domain and range of (f)?
b. what are the domain and range of (g)?
c. identify any asymptotes of (f) and (g).
d. what is ((f \circ g)(x))?
e. what is ((g \circ f)(x))?
f. what is the relationship between (f) and (g)?

1. any questions for me?

Explanation:

Step1: Find domain and range of \(f(x)\)

$$\text{Domain of } f: (-\infty, \infty), \text{ Range of } f: (-\infty, 7)$$

Step2: Find domain and range of \(g(x)\)

$$\text{Domain of } g: (-\infty, 7), \text{ Range of } g: (-\infty, \infty)$$

Step3: Identify asymptotes of \(f\) and \(g\)

$$\lim_{x \to -\infty} f(x) = 7 \implies y = 7; \lim_{x \to 7^-} g(x) = -\infty \implies x = 7$$

Step4: Compute composition \((f \circ g)(x)\)

$$f(g(x)) = 7 - e^{\ln(7-x)} = 7 - (7-x) = x$$

Step5: Compute composition \((g \circ f)(x)\)

$$g(f(x)) = \ln(7 - (7-e^x)) = \ln(e^x) = x$$

Step6: Determine relationship between \(f\) and \(g\)

$$(f \circ g)(x) = (g \circ f)(x) = x \implies f = g^{-1}$$

Answer:

A. Domain of \(f\): \((-\infty, \infty)\); Range of \(f\): \((-\infty, 7)\)
B. Domain of \(g\): \((-\infty, 7)\); Range of \(g\): \((-\infty, \infty)\)
C. Horizontal asymptote of \(f\): \(y = 7\); Vertical asymptote of \(g\): \(x = 7\)
D. \((f \circ g)(x) = x\)
E. \((g \circ f)(x) = x\)
F. \(f\) and \(g\) are inverse functions of each other.