raph the function. Give the domain and range.

f left parenthesis x right parenthesis equals left double bracket negative three halves x right double bracketf(x)=−32x




To graph the function \( f(x) = -\frac{3}{2}x \), we first recognize that this is a linear function. The function represents a straight line with the slope \( -\frac{3}{2} \) and the y-intercept at 0 (since there is no constant term). Now, let's break this down step by step:

### 1. **Slope and Y-intercept**:
   - The **slope** is \( -\frac{3}{2} \), which means that for every 2 units you move to the right (in the positive direction along the x-axis), the graph goes down by 3 units.
   - The **y-intercept** is 0, which means the line passes through the origin, point \( (0, 0) \).

### 2. **Domain**:
   The function is linear, so it is defined for all real numbers. Thus, the domain of \( f(x) \) is:
   \[
   \text{Domain: } (-\infty, \infty)
   \]

### 3. **Range**:
   Since this is a linear function without any restrictions, the y-values can also take any real number. Therefore, the range of \( f(x) \) is:
   \[
   \text{Range: } (-\infty, \infty)
   \]

### 4. **Graph**:
   The graph of \( f(x) = -\frac{3}{2}x \) is a straight line with a downward slope that passes through the origin. Some key points on the graph:
   - At \( x = 0 \), \( f(x) = 0 \) (the origin).
   - At \( x = 2 \), \( f(x) = -3 \) (point \( (2, -3) \)).
   - At \( x = -2 \), \( f(x) = 3 \) (point \( (-2, 3) \)).

These points will help you sketch the line.

Would you like me to plot this graph for you or provide more details? Let me know if you have any questions!

### 5 Related Questions:
1. What happens to the graph if the slope is changed to a positive number?
2. How would the graph change if we added a constant term to the function (e.g., \( f(x) = -\frac{3}{2}x + 1 \))?
3. What is the interpretation of the slope in the context of real-world problems?
4. How do we find the x-intercept and y-intercept of a linear function?
5. How can you determine if two lines are parallel or perpendicular based on their slopes?

### Tip:
When graphing linear functions, always plot at least two points (often the intercepts) and use the slope to sketch the line accurately.

Graph the function. Give the domain and range. f(x) = -\frac{3}{2}x

Learn how to graph the linear function f(x) = -3/2x, including determining the domain and range. This problem focuses on key concepts like slope, y-intercept, and properties of linear functions. Suitable for students in Grades 8-10.

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