Use the given function $$ f $$ to answer parts (a) through (f) below. $$ f(x)=\log (x-4)+2 $$ (c) From the graph, determine the range and any asymptotes of $$ f $$. The range of $$ f $$ is $$ (-\infty, \infty) $$. (Type your answer in interval notation.) Determine the vertical asymptote of $$ f $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. B. The vertical asymptote of $$ f $$ is $$ x=4 $$. There is no vertical asymptote. (Simplify your answer.) (d) Find $$ f^{-1} $$, the inverse of $$ f $$. $$ f^{-1}(x)=10^{x-2}+4 $$ (Simplify your answer.) (e) Find the domain and range of $$ f^{-1} $$. The domain of $$ f^{-1} $$ is $$ \square $$. (Type your answer in interval notation.)

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(c) The function is given by  
$$
f(x)=\log (x-4)+2.
$$  
Since the logarithm is defined for positive arguments, we have  
$$
x-4>0\quad\Longrightarrow\quad x>4.
$$  
Thus, the domain of $$ f $$ is $$(4, \infty)$$. For logarithmic functions of the form $$\log (x-a)$$, the horizontal output covers all real numbers. Therefore, the range of $$ f $$ is  
$$
(-\infty,\infty).
$$

The vertical asymptote occurs where the argument of the logarithm is zero. Setting  
$$
x-4=0 \quad\Longrightarrow\quad x=4,
$$  
we see that the vertical asymptote is at  
$$
x=4.
$$

---

(d) To find the inverse function $$ f^{-1} $$, start with  
$$
y=\log (x-4)+2.
$$  
Interchange $$ x $$ and $$ y $$ to obtain  
$$
x=\log (y-4)+2.
$$  
Subtract 2 from both sides:  
$$
x-2=\log (y-4).
$$  
Now rewrite the logarithmic equation in exponential form (assuming base 10):  
$$
y-4=10^{\,x-2}.
$$  
Finally, solve for $$ y $$:  
$$
y=10^{\,x-2}+4.
$$  
Thus, the inverse function is  
$$
f^{-1}(x)=10^{\,x-2}+4.
$$

---

(e) The domain of $$ f^{-1} $$ is the range of $$ f $$, and the range of $$ f^{-1} $$ is the domain of $$ f $$. Since

- The range of $$ f $$ is $$(-\infty,\infty)$$, the domain of $$ f^{-1} $$ is  
  $$
  (-\infty,\infty).
  $$
- The domain of $$ f $$ is $$(4, \infty)$$, so the range of $$ f^{-1} $$ is  
  $$
  (4,\infty).
  $$

Thus, the domain of $$ f^{-1} $$ is $$(-\infty,\infty)$$ and its range is $$(4,\infty)$$.
(c) The range of $$ f $$ is $$ (-\infty, \infty) $$. The vertical asymptote of $$ f $$ is $$ x=4 $$. (d) The inverse function is $$ f^{-1}(x)=10^{x-2}+4 $$. (e) The domain of $$ f^{-1} $$ is $$ (-\infty, \infty) $$.

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