**Using the function $ f(x) = 3^{x-5} - 4 $ to answer the following questions:**(a) **Determine the equation of the asymptote of $ f(x) $:**\[y = -4\](b) **Determine the domain of $ f(x) $ in interval notation:**\[(-\infty, \infty)\](c) **Determine the range of $ f(x) $ in interval notation:**\[(-4, \infty)\]

Answered: User the function f(x) = 3"-5 - 4 to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Step 1

(a)

The given function $f (x) = 3^{x - 5} - 4$ is an exponential function.

It is known that, the exponential function generally has no vertical asymptotes.

Note that, the horizontal asymptote of the function y=f(x) is the line y=L, if either $\lim_{x \to \infty} f (x) = L o r \lim_{x \to - \infty} f (x) = L$ .

Obtain the value of $\lim_{x \to \infty} f (x) = L$ .

$\begin{array}{rcl} \lim_{x \to \infty} f (x) & = & \lim_{x \to \infty} (3^{x - 5} - 4) \\ = & \lim_{x \to \infty} (3^{x - 5}) - \lim_{x \to \infty} (4) \\ = & \infty - 4 \\ = & \infty \end{array}$

Obtain the value of $\lim_{x \to - \infty} f (x) = L$ .

$\begin{array}{rcl} \lim_{x \to - \infty} f (x) & = & \lim_{x \to - \infty} (3^{x - 5} - 4) \\ = & \lim_{x \to - \infty} (3^{x - 5}) - \lim_{x \to - \infty} (4) \\ = & 0 - 4 \\ = & - 4 \end{array}$

Therefore, the horizontal asymptote of the given function is $y = - 4$

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