To analyze the function \( n(x) = 5x^2 - 20x + 12 \), we will explore its domain and range.
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Domain: The function is a quadratic function, and quadratic functions are defined for all real numbers. Therefore, the domain of \( n(x) \) is: \[ \text{Domain: } (-\infty, \infty) \]
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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: \[ \text{Range: } n(x) \geq -8 \]
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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 \( n(x) \geq -8 \).