https://file.aiquickdraw.com/mathbot/file/1729737749250xyxqxff4.jpg

Let's work through this step by step based on the image provided.

### (a) **Finding the Domain:**
The function is defined in three parts:

- \( f(x) = \frac{2}{x^2} \) if \( 0 \leq x \leq 4 \)
- \( f(x) = x + 12 \) if \( 4 < x \leq 7 \)
- \( f(x) = 19 \) if \( x > 7 \)

The domain consists of all \( x \)-values where the function is defined. 
- For \( f(x) = \frac{2}{x^2} \), it is defined for \( 0 \leq x \leq 4 \).
- For \( f(x) = x + 12 \), it's defined for \( 4 < x \leq 7 \).
- For \( f(x) = 19 \), it's defined for \( x > 7 \).

Thus, the **domain** is \( [0, \infty) \).

### (b) **Finding Intercepts:**

- **Y-intercept**:
   The y-intercept occurs where \( x = 0 \). Using the first part of the function:
   \[
   f(0) = \frac{2}{0^2} = \infty
   \]
   This implies there is no y-intercept, because \( f(0) \) is undefined.

- **X-intercept**:
   The x-intercept occurs where \( f(x) = 0 \).
   1. For \( f(x) = \frac{2}{x^2} \), it can never equal zero.
   2. For \( f(x) = x + 12 \), set \( x + 12 = 0 \):
      \[
      x = -12
      \]
      But this is not in the domain of the function.
   3. For \( f(x) = 19 \), it can never equal zero.

Thus, there is **no x-intercept**.

### (c) **Graphing the Function:**
You need to select the correct graph based on the function. Here's how the function behaves in the three intervals:

1. **For \( 0 \leq x \leq 4 \)**, the function \( f(x) = \frac{2}{x^2} \) decreases as \( x \) increases.
2. **For \( 4 < x \leq 7 \)**, the function \( f(x) = x + 12 \) is a straight line with slope 1.
3. **For \( x > 7 \)**, the function is constant, \( f(x) = 19 \).

Based on this behavior, the correct graph will reflect these characteristics.

### (d) **Finding the Range:**
- For \( 0 \leq x \leq 4 \), \( f(x) = \frac{2}{x^2} \) decreases from \( \frac{2}{4^2} = \frac{2}{16} \) to \( \infty \).
- For \( 4 < x \leq 7 \), \( f(x) = x + 12 \), so it increases from 16 to 19.
- For \( x > 7 \), \( f(x) = 19 \).

Thus, the **range** is \( (0, 19] \).

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Would you like any further details on any part of this? 

Here are five related questions to explore:
1. What are the conditions for the existence of intercepts in different types of functions?
2. How do piecewise functions affect continuity at boundary points?
3. What are typical methods to find the range of piecewise functions?
4. How can graphing help in visualizing and confirming function properties?
5. What happens if the domain of a function includes a point where it's undefined?

**Tip**: For piecewise functions, always check the behavior at the boundaries, as they can sometimes indicate discontinuities or breaks.

Find the domain, intercepts, graph, and range of a piecewise function f(x) with the following rules: f(x) = 2/x^2 for 0 ≤ x ≤ 4, f(x) = x + 12 for 4 < x ≤ 7, f(x) = 19 for x > 7.

Learn how to analyze a piecewise function step by step. This example covers finding the domain, intercepts, graph, and range of the function f(x), which is defined in three parts: f(x) = 2/x^2 for 0 ≤ x ≤ 4, f(x) = x + 12 for 4 < x ≤ 7, and f(x) = 19 for x > 7. Suitable for pre-calculus students.

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