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To find the average value of the function \( f(x) \) on the interval \([0, 32]\), we use the formula for the average value of a function on an interval:

\[
f_{\text{ave}} = \frac{1}{b - a} \int_a^b f(x) \, dx
\]

Where:
- \( a = 0 \)
- \( b = 32 \)

From the graph, \( f(x) \) is a piecewise linear function. We need to find the area under the graph between \( x = 0 \) and \( x = 32 \). Then, divide that area by the length of the interval, which is \( 32 - 0 = 32 \).

### Step 1: Break down the graph into geometric shapes.
The function is a series of line segments forming triangles and trapezoids. We calculate the area under each segment.

1. From \( x = 0 \) to \( x = 8 \): 
   - This forms a triangle with base \( 8 \) and height \( 2 \).
   - Area of triangle = \( \frac{1}{2} \times 8 \times 2 = 8 \).

2. From \( x = 8 \) to \( x = 16 \):
   - This forms a trapezoid with parallel sides of length 0 and 2, and a height of 8.
   - Area of trapezoid = \( \frac{1}{2} \times (2 + 0) \times 8 = 8 \).

3. From \( x = 16 \) to \( x = 24 \):
   - This forms another trapezoid with parallel sides 2 and 0, and a height of 8.
   - Area of trapezoid = \( \frac{1}{2} \times (2 + 0) \times 8 = 8 \).

4. From \( x = 24 \) to \( x = 32 \):
   - This forms a triangle with base \( 8 \) and height \( 4 \).
   - Area of triangle = \( \frac{1}{2} \times 8 \times 4 = 16 \).

### Step 2: Calculate the total area under the curve.
Total area = \( 8 + 8 + 8 + 16 = 40 \).

### Step 3: Apply the average value formula.
Now, we apply the formula:

\[
f_{\text{ave}} = \frac{1}{32 - 0} \times 40 = \frac{40}{32} = 1.25
\]

Thus, the average value of \( f \) on the interval \([0, 32]\) is \( f_{\text{ave}} = 1.25 \).

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Do you want more details or have any questions?

Here are 5 related questions:
1. How is the average value of a function on an interval generally interpreted?
2. What is the significance of the area under a curve in calculus?
3. Can the average value of a function be negative? If so, when?
4. How would you approach finding the average value if the function was not piecewise linear?
5. What does the graph tell us about the general behavior of the function on the interval?

**Tip:** Breaking down graphs into simple geometric shapes like triangles and trapezoids makes it much easier to compute areas manually.

Learn how to find the average value of a function f(x) on the interval [0, 32] by calculating the area under a piecewise linear function. The problem involves key calculus concepts such as the average value of a function and the use of geometric shapes like triangles and trapezoids. Suitable for high school students in Grades 10-12.

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