Question

Which statement about the function n(x) = 5x ^ 2 - 20x + 12 is true?
A The domain of the function is x >= 2
The range of the function is n(x) >= - 8
The domain of the function is x >= 0
⑥ The range of the function is n(x) <= 12

Answer 1

To analyze the function \( n(x) = 5x^2 - 20x + 12 \), we will explore its domain and range.

1. **Domain**: The function is a quadratic function, and quadratic functions are defined for all real numbers. Therefore, the domain of \( n(x) \) is:
\[
\text{Domain: } (-\infty, \infty)
\]

2. **Range**: To find the range, we need to determine the vertex of the quadratic function since it opens upwards (the coefficient of \( x^2 \) is positive).

The vertex \( x \)-coordinate can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
where \( a = 5 \) and \( b = -20 \).

\[
x = -\frac{-20}{2 \cdot 5} = \frac{20}{10} = 2
\]

Now, we evaluate \( n(x) \) at \( x = 2 \):
\[
n(2) = 5(2^2) - 20(2) + 12 = 20 - 40 + 12 = -8
\]

Since the parabola opens upwards and the vertex is the minimum point, the range of the function is:
\[
\text{Range: } n(x) \geq -8
\]

3. **Evaluating the statements**:
- A: The domain of the function is \( x \geq 2 \) (False)
- B: The range of the function is \( n(x) \geq -8 \) (True)
- C: The domain of the function is \( x \geq 0 \) (False)
- D: The range of the function is \( n(x) \leq 12 \) (False)

Thus, the only true statement is:
**The range of the function is \( n(x) \geq -8 \)**.