Question

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?
2. please respond to one of your fellow students posts.

Explanation:

Step 1: Find domain and range of \(f(x)\)

$$\text{Domain of } f(x) = 7 - e^x \text{ is } (-\infty, \infty). \text{ Since } e^x > 0, \text{ the range is } (-\infty, 7).$$

Step 2: Find domain and range of \(g(x)\)

$$\text{For } g(x) = \ln(7 - x), \text{ we need } 7 - x > 0 \implies x < 7. \text{ Domain is } (-\infty, 7). \text{ Range is } (-\infty, \infty).$$

Step 3: Identify asymptotes of \(f\) and \(g\)

$$\text{As } x \to -\infty, f(x) \to 7, \text{ so horizontal asymptote is } y = 7. \text{ As } x \to 7^-, g(x) \to -\infty, \text{ so vertical asymptote is } x = 7.$$

Step 4: Compute composition \((f \circ g)(x)\)

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

Step 5: Compute composition \((g \circ f)(x)\)

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

Step 6: Determine relationship between \(f\) and \(g\)

$$\text{Since } (f \circ g)(x) = x \text{ and } (g \circ f)(x) = x, f \text{ and } g \text{ are inverse functions.}$$

Answer:

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