Consider the following function.

*f*(*x*) = ln(*x*) + 6

(a) What are the domain and range of *f*? (Enter your answers using interval notation.)

domain

range

(b) What is the *x*-intercept of the graph of *f*?

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 \( x \).

Consider the following function. f(x) = ln(x) + 6 (a) What are the domain and range of f? (Enter your answers using interval notation.) domain range (b) What is the x-intercept of the graph of f?

Explore the domain, range, and x-intercept of the logarithmic function f(x) = ln(x) + 6. This problem demonstrates key concepts such as the domain and range of logarithmic functions and how to find the x-intercept using properties of natural logarithms. Suitable for students in Grades 10-12.

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