Use the given function $$ f $$ to answer parts (a) through (f) below. $$ f(x)=\log (x-4)+4 $$

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**(a) Domain**

The function is given by  
$$
f(x)=\log (x-4)+4.
$$  
Since the logarithm function is only defined for positive arguments, we require  
$$
x-4>0.
$$  
Solving for $$ x $$:  
$$
x>4.
$$  
Thus, the domain is  
$$
(4, \infty).
$$

---

**(b) Range**

The logarithmic function, regardless of shifts, has a range of all real numbers. Since  
$$
\log (x-4)
$$  
can take any real value as $$ x-4 $$ ranges over $$ (0,\infty) $$, adding 4 only shifts the output vertically. Therefore, the range is  
$$
(-\infty, \infty).
$$

---

**(c) $$ x $$-Intercept**

The $$ x $$-intercept occurs where $$ f(x)=0 $$. Setting the function equal to zero gives:  
$$
\log (x-4)+4=0.
$$  
Subtract 4 from both sides:  
$$
\log (x-4)=-4.
$$
Exponentiating both sides (assuming the logarithm is base 10) gives:  
$$
x-4=10^{-4}.
$$
Thus,  
$$
x=4+10^{-4}.
$$
So the $$ x $$-intercept is  
$$
\left(4+10^{-4},\;0\right).
$$

---

**(d) $$ y $$-Intercept**

The $$ y $$-intercept occurs where $$ x=0 $$. However, note that the domain of the function is $$ x>4 $$. Since $$ x=0 $$ is not in the domain, there is no $$ y $$-intercept.

---

**(e) Vertical Asymptote**

Vertical asymptotes for a logarithmic function occur where the argument of the logarithm is zero. Set  
$$
x-4=0,
$$
which gives  
$$
x=4.
$$
Therefore, there is a vertical asymptote at  
$$
x=4.
$$

---

**(f) Horizontal Asymptote**

Examine the behavior of $$ f(x) $$ as $$ x\to\infty $$:  
$$
f(x)=\log (x-4)+4.
$$
The logarithm function increases without bound (albeit slowly) as its argument approaches infinity. Hence,  
$$
\lim_{x\to\infty} f(x)=\infty.
$$
Thus, there is no horizontal asymptote.
**(a) Domain:** $$ x > 4 $$ **(b) Range:** All real numbers **(c) $$ x $$-Intercept:** $$ (4.0001, 0) $$ **(d) $$ y $$-Intercept:** None **(e) Vertical Asymptote:** $$ x = 4 $$ **(f) Horizontal Asymptote:** None

Use the given function \( f \) to answer parts (a) through (f) below. \[ f(x)=\log (x-4)+4 \]

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