Let's analyze the function $f(x) = \ln(x) + 6$ in parts:

(a) Domain and Range of $f(x)$

Domain:

The function involves the natural logarithm, $\ln(x)$, which is only defined when $x > 0$. Therefore, the domain is all positive real numbers. In interval notation, this is:

$\text{Domain}: (0, \infty)$

Range:

The logarithmic function $\ln(x)$ takes values from $-\infty$ to $\infty$ as $x$ increases from 0 to infinity. Since the function adds 6 to $\ln(x)$, the range is also shifted up by 6, meaning:

$\text{Range}: (-\infty + 6, \infty + 6) = (-\infty, \infty)$

Thus, the range is all real numbers:

$\text{Range}: (-\infty, \infty)$

(b) $x$-intercept of the graph of $f$

To find the $x$-intercept, we set $f(x) = 0$ and solve for $x$:

$f(x) = \ln(x) + 6 = 0$

$\ln(x) = -6$

Now, solve for $x$ by exponentiating both sides:

$x = e^{-6}$

Thus, the $x$-intercept is at $x = e^{-6}$.

Summary of answers:

(a)

• Domain: $(0, \infty)$
• Range: $(-\infty, \infty)$

(b)

• $x$-intercept: $x = e^{-6}$

Would you like more details on any part of the solution?

Here are 5 related questions:

1. What is the behavior of $f(x)$ as $x \to 0^+$?
2. How would the graph of $f(x) = \ln(x) + 6$ compare to $f(x) = \ln(x)$?
3. What happens to the range of $f(x)$ if we change the constant from 6 to 2?
4. What is the derivative of $f(x) = \ln(x) + 6$?
5. How does the $y$-intercept compare between logarithmic functions with different constants?

Tip: For functions involving logarithms, always check the domain carefully, as logarithms are only defined for positive values of $x$.