The function y = x^9 is a polynomial function, which means it is continuous and defined for all real numbers. Let’s examine its domain and range in detail.
Domain:
The domain of a function is the set of all possible input values (x-values). For the function y = x^9, you can input any real number for x, including positive numbers, negative numbers, and zero. Therefore, the domain of this function is:
- Domain: All real numbers (or in interval notation:
(−∞, +∞))
Range:
The range of a function is the set of all possible output values (y-values). Since y = x^9 is an odd-degree polynomial, we can analyze its behavior:
- As x approaches positive infinity (
x → +∞), y also approaches positive infinity (y → +∞). - As x approaches negative infinity (
x → -∞), y approaches negative infinity (y → -∞).
This means the function can output any real number, covering all possible y-values. Thus, the range of this function is:
- Range: All real numbers (or in interval notation:
(−∞, +∞))
In summary, for the function y = x^9, both the domain and range are:
- Domain:
(−∞, +∞) - Range:
(−∞, +∞)
This means you can input any real number into the function, and it can output any real number as well.