Consider the function $$ f(x)=-2 x^{2}+16 x-9 $$ a. Determine, without graphing, whether the function has a minimum value or a maximum value a. Find the minimum or maximum value and determine where if occurs c. Identify the function's domain and its range a. The function has a maximum value b. The minimumimaximum value is 23 it occurs at $$ x=4 $$ c. The domain of $$ f $$ is 23 (Type your answer in interval notation )

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Let's carefully analyze the function $$ f(x) = -2x^2 + 16x - 9 $$ and address each part of the problem step by step. I'll also point out and correct any discrepancies in your initial answers.

### a. Determine whether the function has a minimum or a maximum value

**Your Answer:**
> The function has a maximum value.

**Evaluation:**
Your answer is **correct**.

**Explanation:**
The function $$ f(x) = -2x^2 + 16x - 9 $$ is a quadratic function of the form $$ f(x) = ax^2 + bx + c $$, where $$ a = -2 $$. Since the coefficient $$ a $$ is negative, the parabola opens downward, which means the function has a **maximum** value.

---

### b. Find the minimum or maximum value and determine where it occurs

**Your Answer:**
> The minimumimaximum value is 23 it occurs at $$ x = 4 $$.

**Evaluation:**
There's a slight confusion in your terminology ("minimumimaximum"). Additionally, we need to verify the maximum value and the point where it occurs.

**Correct Answer:**
The function has a **maximum** value of **23**, and it occurs at $$ x = 4 $$.

**Verification:**

To find the maximum value and where it occurs, we can use the vertex formula for a quadratic function:

$$
x = -\frac{b}{2a}
$$

Given:
$$
a = -2, \quad b = 16
$$

Calculate $$ x $$:

$$
x = -\frac{16}{2 \times (-2)} = -\frac{16}{-4} = 4
$$

Now, find $$ f(4) $$:

$$
f(4) = -2(4)^2 + 16(4) - 9 = -2(16) + 64 - 9 = -32 + 64 - 9 = 23
$$

So, the maximum value is **23**, and it occurs at $$ x = 4 $$.

---

### c. Identify the function's domain and its range

**Your Answer:**
> The domain of $$ f $$ is 23 (Type your answer in interval notation )

**Evaluation:**
There seems to be a misunderstanding here. The **domain** of a function refers to all possible input values (x-values), while the **range** refers to all possible output values (f(x)-values). Additionally, the number "23" refers to the maximum value of the function, not its domain.

**Correct Answer:**

- **Domain:** The domain of $$ f(x) = -2x^2 + 16x - 9 $$ is **all real numbers**.

$$
\text{Domain} = (-\infty, \infty)
$$

- **Range:** Since the parabola opens downward and its maximum value is 23, the range includes all real numbers less than or equal to 23.

$$
\text{Range} = (-\infty, 23]
$$

**Explanation:**

- **Domain:** Quadratic functions are defined for all real numbers unless there are restrictions (like division by zero or square roots of negative numbers). In this case, there are no such restrictions.

- **Range:** The maximum value of the function is 23 (as found in part b). Since the parabola opens downward, the function values decrease without bound from the vertex. Therefore, the range is all real numbers **up to and including** 23.

---

### Summary of Correct Answers

a. **The function has a maximum value.**

b. **The maximum value is 23, and it occurs at $$ x = 4 $$.**

c. 
- **Domain:** $$ (-\infty, \infty) $$
- **Range:** $$ (-\infty, 23] $$

If you have any further questions or need additional clarification, feel free to ask!
The function $$ f(x) = -2x^2 + 16x - 9 $$ has a maximum value of 23 at $$ x = 4 $$. Its domain is all real numbers, and its range is all real numbers less than or equal to 23.

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