Question

use the given function f to answer parts (a) through (f).

f(x)=e^{x + 3}+1

(c) from the graph, determine the range and any asymptotes of f.

the range of f is (1,\infty).

(type your answer in interval notation.)

determine the horizontal asymptote of f, if it exists. select the correct choice and, if necessary, fill in the answer box to complete your choice.

a. the horizontal asymptote of f is y = 1.

(simplify your answer.)

b. there is no horizontal asymptote.

(d) find f^{-1}, the inverse of f.

f^{-1}(x)=\square (simplify your answer.)

Explanation:

Step1: Replace \(f(x)\) with \(y\)

\(y = e^{x + 3}+1\)

Step2: Swap \(x\) and \(y\)

\(x=e^{y + 3}+1\)

Step3: Solve for \(y\)

Subtract \(1\) from both sides: \(x - 1=e^{y + 3}\)
Take the natural logarithm of both sides: \(\ln(x - 1)=\ln(e^{y + 3})\)
Since \(\ln(e^{a})=a\), we have \(\ln(x - 1)=y + 3\)
Subtract \(3\) from both sides: \(y=\ln(x - 1)-3\)

Answer:

\(f^{-1}(x)=\ln(x - 1)-3\)