To determine the domain and range of the polynomial function \( y = -x^3 + 3x^2 + x - 3 \):
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Domain: The domain of any polynomial function is all real numbers since there are no restrictions on the input values. Therefore, the domain is \( (-\infty, \infty) \).
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Range: To find the range, we need to analyze the behavior of the function. This polynomial is a cubic function, and since it has a negative leading coefficient (\(-x^3\)), the function will fall towards negative infinity as \( x \) approaches positive infinity and rise towards positive infinity as \( x \) approaches negative infinity.
To better understand the range, we can find critical points by taking the derivative and setting it to zero.
First, compute the derivative: \[ y' = -3x^2 + 6x + 1 \]
Setting the derivative to zero to find critical points: \[ -3x^2 + 6x + 1 = 0 \] Dividing by -1 gives: \[ 3x^2 - 6x - 1 = 0 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot (-1)}}{2 \cdot 3} = \frac{6 \pm \sqrt{36 + 12}}{6} = \frac{6 \pm \sqrt{48}}{6} = \frac{6 \pm 4\sqrt{3}}{6} = 1 \pm \frac{2\sqrt{3}}{3} \]
Finding the y-values at these critical points helps establish the maximum or minimum for real outputs. However, identifying that the polynomial must tend toward negative infinity at both ends implies the range covers all real numbers.
Thus, the range of the function is \( (-\infty, \infty) \).
So the domain and range of the polynomial \( y = -x^3 + 3x^2 + x - 3 \) are:
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, \infty) \)
The correct response is Domain: (−∞,∞); Range: (−∞,∞).