Consider the function on the interval -3<x<4
.

f(x)= -1/3x+7
What are the domain, range, minimum value, and maximum value of the function over the interval?

Enter the numbers in the boxes.

Let's analyze the function \( f(x) = -\frac{1}{3}x + 7 \) over the interval \( -3 < x < 4 \).

### 1. Domain:
The domain is given by the interval \( -3 < x < 4 \).

### 2. Range:
The range can be determined by evaluating the function at the endpoints of the interval.

- **At \( x = -3 \)**:
\[
f(-3) = -\frac{1}{3}(-3) + 7 = 1 + 7 = 8
\]

- **At \( x = 4 \)**:
\[
f(4) = -\frac{1}{3}(4) + 7 = -\frac{4}{3} + 7 = 7 - \frac{4}{3} = \frac{21 - 4}{3} = \frac{17}{3} \approx 5.67
\]

Since the function is linear and decreasing, the maximum value occurs at \( x = -3 \) and the minimum value occurs at \( x = 4 \).

### 3. Maximum value:
The maximum value is \( f(-3) = 8 \).

### 4. Minimum value:
The minimum value is \( f(4) = \frac{17}{3} \approx 5.67 \).

### 5. Range:
Since the function decreases over the interval, the range of \( f(x) \) is \( \left(\frac{17}{3}, 8\right) \).

### Summary:
- **Domain**: \( -3 < x < 4 \)
- **Range**: \( \frac{17}{3} < y < 8 \) 
- **Minimum Value**: \( \frac{17}{3} \approx 5.67 \)
- **Maximum Value**: \( 8 \)

Would you like a more detailed explanation or have any questions?

Here are five related questions:
1. What is the slope of the function \( f(x) \)?
2. How would the range change if the interval was \( -2 < x < 3 \)?
3. Can this linear function ever have a minimum value of 10 over any interval?
4. How does the function behave as \( x \) approaches the endpoints of the domain?
5. What would be the effect of changing the constant term in the function?

**Tip:** Always check the behavior of the function at the endpoints of the interval to determine the range.

Explore the domain, range, minimum value, and maximum value of a linear function f(x) = -1/3x + 7 over the interval -3 < x < 4. Learn how to determine these values and understand the behavior of linear functions in this interval.

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