https://file.aiquickdraw.com/mathbot/file/1726722985859fo36dhxa.jpg

The function given in the image is \( f(x) = \ln(|x - 7|) \), and we are asked to find where the function is discontinuous and identify the type of discontinuities.

### Step 1: Analyze the logarithmic function
The natural logarithm function \( \ln(x) \) is only defined for \( x > 0 \). Therefore, for the function \( \ln(|x - 7|) \), the argument \( |x - 7| \) must also be greater than zero. This means:
\[
|x - 7| > 0 \implies x \neq 7
\]
So, \( f(x) \) is undefined at \( x = 7 \). Additionally, we must check the domain of the absolute value term \( |x - 7| \). Since the absolute value is non-negative, the function is defined as long as \( x \neq 7 \), and thus there is a discontinuity at \( x = 7 \).

### Step 2: Identify the type of discontinuity
At \( x = 7 \), we need to classify the discontinuity:
- The function approaches negative infinity as \( x \) approaches 7 from the left and right (since \( \ln(0^+) = -\infty \)).
- This indicates that the discontinuity at \( x = 7 \) is an **infinite discontinuity**.

### Final Answer:
- \( f(x) \) is discontinuous at \( x = 7 \).
- The type of discontinuity is **infinite**.

Do you want more details or have any questions?

### Follow-up questions:
1. How would the graph of \( f(x) = \ln(|x - 7|) \) look around the discontinuity?
2. How does the absolute value impact the function compared to \( \ln(x - 7) \)?
3. What other types of discontinuities could exist in general functions?
4. How does one handle piecewise functions to determine discontinuities?
5. Can a function have more than one type of discontinuity at the same point?

**Tip:** For any function involving \( \ln(x) \), always check if the argument is positive to determine where the function is defined.

Consider the function f(x) = ln(|x - 7|). Where are the discontinuities located? Identify the type or types of discontinuity.

Explore the discontinuities in the logarithmic function f(x) = ln(|x - 7|). This problem involves the domain of logarithmic functions and classifies the discontinuity at x = 7 as an infinite discontinuity. Suitable for students studying Pre-Calculus or advanced algebra topics.

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