Answered

$f(x)=-(x+3)(x-1)$ is shown.



What is true about the domain and range of the function?


AThe domain is all real numbers such that $-3 \leq x \leq 1$, and the range is all real numbers less than or equal to 4.
BThe domain is all real numbers less than or equal to 4, and the range is all real numbers such that $-3 \leq x \leq 1$.
CThe domain is all real numbers less than or equal to 4, and the range is all real numbers.
DThe domain is all real numbers, and the range is all real numbers less than or equal to 4.

Verified Answer :

  • The function \(f(x) = -(x+3)(x-1)\) is a polynomial, so its domain is all real numbers.
  • As a downward-opening quadratic, its maximum value is at the vertex, found at \(x = -1\).
  • Evaluating at the vertex gives a maximum value of \(f(-1) = 4\).
  • The domain is all real numbers, and the range is all real numbers less than or equal to 4.
  • The correct statement is: \(\boxed{\text{The domain is all real numbers, and the range is all real numbers less than or equal to 4.}}\)

Explanation

  1. Analyze the Domain

The function is given by \(f(x) = -(x+3)(x-1)\). This is a quadratic function, which is a type of polynomial. The domain of any polynomial function is the set of all real numbers, since you can plug in any real number into the polynomial expression and obtain a valid output. Thus, the domain is \((-\infty, \infty)\).

  1. Determine the Range and Conclusion

To find the range, we identify the shape and extreme value of the function. The function is a downward-opening parabola because the leading coefficient (the coefficient of \(x^2\) when expanded) is negative: \(f(x) = -(x^2 + 2x - 3) = -x^2 - 2x + 3\). The vertex of a parabola \(f(x) = ax^2 + bx + c\) is located at \(x = -\frac{b}{2a}\). Here, \(a = -1\) and \(b = -2\), so the x-coordinate of the vertex is \(x = -\frac{-2}{2(-1)} = \frac{2}{-2} = -1\). Alternatively, the vertex x-coordinate is the midpoint of the roots \(x = -3\) and \(x = 1\), which is \(\frac{-3+1}{2} = -1\). Evaluating the function at \(x = -1\):
\[f(-1) = -(-1+3)(-1-1) = -(2)(-2) = 4.\]
Since the parabola opens downward, the function reaches its maximum value of \(4\) at the vertex. The range is therefore all real numbers less than or equal to 4, denoted as \((-\infty, 4]\).

Examples

In physics, if you throw a ball into the air, its height over time is often described by a quadratic function. Just as this function has a maximum height that it cannot exceed (the vertex of the parabola), the path of the ball is restricted by the limits of the physical environment, demonstrating how domains and ranges help us understand the constraints of real-world motion. Understanding these concepts allows scientists and engineers to predict the maximum peak of a projectile's flight or the minimum energy required for a mechanical process.