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Answer:
Understanding Domain and Range
Before we dive into the specific problems, let's clarify what domain and range mean:
* Domain: The set of all possible input values (x-values) for a function.
* Range: The set of all possible output values (y-values) for a function.
Problem 1: y = -x²
Domain:
* Since we can square any real number, the domain for this function is all real numbers.
* Domain: (-∞, ∞) or all real numbers
Range:
* The square of any real number is always non-negative. However, the negative sign in front of x² flips the parabola upside down. Therefore, the output (y) will always be less than or equal to 0.
* Range: (-∞, 0] or y ≤ 0
Problem 2: y = 1 - x²
Domain:
* Similar to the first problem, we can square any real number.
* Domain: (-∞, ∞) or all real numbers
Range:
* The maximum value of -x² is 0. Therefore, the maximum value of 1 - x² is 1. Since the parabola opens downward, all other y-values will be less than 1.
* Range: (-∞, 1] or y ≤ 1
In summary:
| Function | Domain | Range |
|---|---|---|
| y = -x² | (-∞, ∞) | (-∞, 0] |
| y = 1 - x² | (-∞, ∞) | (-∞, 1] |
Note: The parentheses () indicate that the endpoint is not included, while the square bracket [] indicates that the endpoint is included.