The function you provided is \( f(x) = \frac{2}{3}x + 1 \). This is a linear function of the form \( f(x) = mx + b \), where:

- \( m = \frac{2}{3} \) is the slope of the line.
- \( b = 1 \) is the y-intercept of the line, which is the point where the line crosses the y-axis.

### Key Features of the Function:

1. **Slope**: The slope \( \frac{2}{3} \) indicates that for every increase of 1 unit in \( x \), the value of \( f(x) \) increases by \( \frac{2}{3} \). This represents a gradual upward incline.

2. **Y-Intercept**: The line crosses the y-axis at the point (0, 1).

3. **Graph**: The graph is a straight line with a positive slope, slanting upwards from left to right.

4. **Domain and Range**:
   - **Domain**: All real numbers (\( x \in \mathbb{R} \)).
   - **Range**: All real numbers (\( f(x) \in \mathbb{R} \)).

5. **Behavior**: The function is increasing since the slope is positive.

Would you like a detailed explanation of any aspect, or perhaps a graph? 

### Related Questions:
1. What is the x-intercept of the function \( f(x) = \frac{2}{3}x + 1 \)?
2. How does changing the slope affect the graph of a linear function?
3. How would the function change if the y-intercept was different?
4. Can you determine the rate of change between any two points on this line?
5. What is the significance of the slope in real-world applications?

**Tip:** Always remember, the slope tells you how steep a line is, and the y-intercept shows where the line crosses the y-axis!

Explore the linear function f(x) = 2/3x + 1, where the slope is 2/3 and the y-intercept is 1. This problem provides a detailed analysis of key algebraic concepts, including the slope-intercept form. Perfect for students in Grades 8-10.

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