The function given is:

\[
f(x) = \log_{\frac{3}{2}}(x - 3) + 1
\]

This is a logarithmic function with a few transformations applied. Let's break it down:

### 1. **Logarithmic Base**:
   - The logarithm has a base of \(\frac{3}{2}\), which means it measures the power to which \(\frac{3}{2}\) must be raised to obtain the argument.

### 2. **Inside the Logarithm**:
   - The expression inside the logarithm is \(x - 3\). This indicates a horizontal shift of the logarithmic function to the right by 3 units. 
   - The logarithm is defined only for positive arguments, so \(x - 3 > 0\), meaning \(x > 3\). The domain of \(f(x)\) is therefore \(x > 3\).

### 3. **Vertical Shift**:
   - The \(+1\) outside the logarithm indicates a vertical shift of the entire graph upwards by 1 unit.

### **Summary of Transformations**:
- **Horizontal shift**: Right by 3 units.
- **Vertical shift**: Up by 1 unit.
- **Base of the logarithm**: \(\frac{3}{2}\), meaning it is an increasing logarithm (since \(\frac{3}{2} > 1\)).

### **Domain**:
\[
\text{Domain: } x > 3
\]

### **Range**:
\[
\text{Range: } (-\infty, \infty)
\]

Would you like to explore more details or have any questions about this function?

### Related Questions:
1. What happens to the graph of \(f(x)\) if we change the base to \(2\)?
2. How would the graph change if the logarithmic base was \(\frac{1}{2}\) instead of \(\frac{3}{2}\)?
3. What is the behavior of \(f(x)\) as \(x\) approaches \(3\) from the right?
4. How would you find the x-intercept of the function \(f(x)\)?
5. Can you determine the inverse of this function?

### Tip:
When dealing with logarithms, always pay attention to the domain of the function, as logarithms are only defined for positive arguments.

Explore the logarithmic function f(x) = log base 3/2 of (x - 3) + 1 with detailed explanations of its transformations. Learn about the logarithmic base, horizontal and vertical shifts, domain, and range. Suitable for high school students studying logarithmic functions.

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