Which of the following is NOT a key feature of the function h(x)? h of x equals x squared minus 5 when 0 is less than or equal to x is less than or equal to 3 and negative log base 3 of x when x is greater than 3. The domain of h(x) is [0, ∞). The range of h(x) is (–∞, ∞). The y-intercept of h(x) is (0, –5). The end behavior of h(x) is as x → ∞, h(x) → –∞.

To determine which statement is **not** a key feature of the function \( h(x) \), let's analyze the given information and each statement:

### Given Function Definition:
1. \( h(x) = x^2 - 5 \) when \( 0 \leq x \leq 3 \)
2. \( h(x) = -\log_3(x) \) when \( x > 3 \)

### Analysis of the Statements:

1. **Domain of \( h(x) \):** 
   - The first part \( x^2 - 5 \) is defined for \( 0 \leq x \leq 3 \).
   - The second part \( -\log_3(x) \) is defined for \( x > 3 \).
   - Combining these, the domain of \( h(x) \) is \( [0, \infty) \).
   - **This statement is correct.**

2. **Range of \( h(x) \):**
   - For \( x^2 - 5 \), when \( 0 \leq x \leq 3 \), the output ranges from \( -5 \) (at \( x = 0 \)) to \( 4 \) (at \( x = 3 \)).
   - For \( -\log_3(x) \), when \( x > 3 \), the function will produce negative values decreasing without bound as \( x \) increases.
   - Thus, the range of \( h(x) \) is \( (-\infty, 4] \).
   - **This contradicts the statement that the range is \( (-\infty, \infty) \). This statement is incorrect.**

3. **The y-intercept of \( h(x) \):**
   - The y-intercept occurs when \( x = 0 \). Here, \( h(0) = 0^2 - 5 = -5 \).
   - The y-intercept is indeed \( (0, -5) \).
   - **This statement is correct.**

4. **End behavior of \( h(x) \) as \( x \to \infty \):**
   - As \( x \to \infty \), \( h(x) = -\log_3(x) \) and \( \log(x) \) grows unbounded, so \( -\log_3(x) \) approaches \( -\infty \).
   - **This statement is correct.**

### Conclusion:
The incorrect statement is that **the range of \( h(x) \) is \( (-\infty, \infty) \)**. The correct range is \( (-\infty, 4] \).

### Summary:
The feature that is NOT true about the function \( h(x) \) is:

**The range of \( h(x) \) is \( (-\infty, \infty) \).**

Would you like more details or have any questions? Here are five related questions:

1. How do you determine the range of a piecewise function?
2. What is the importance of identifying the domain and range in functions?
3. Can logarithmic functions have a range of \( (-\infty, \infty) \)?
4. How does end behavior influence the analysis of functions?
5. Why is it important to verify each feature of a function when analyzing it?

**Tip:** Always graph a function if possible, as it helps visually confirm the domain, range, intercepts, and end behavior.

Explore the properties of the function h(x) = x^2 - 5 for 0 ≤ x ≤ 3 and h(x) = -log₃(x) for x > 3. Understand its domain, range, y-intercept, and end behavior. Suitable for high school students.

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