The function f(x) = x6 is a polynomial function, and understanding its domain and range is essential for analyzing its behavior.
Domain: The domain of a function refers to all the possible input values (x-values) that can be used in the function. For polynomial functions like f(x) = x6, there are no restrictions on the input, meaning you can plug in any real number. Therefore, the domain of f(x) is:
- Domain: All real numbers, which can be expressed in interval notation as (-∞, +∞).
Range: The range of a function consists of all possible output values (y-values) that the function can produce. Since f(x) = x6 is an even-powered polynomial, it will always yield non-negative results regardless of the input value of x. This is because raising any real number to an even power results in a non-negative value.
As x approaches positive or negative infinity, f(x) also approaches positive infinity, and the minimum value occurs at x = 0, where:
f(0) = 06 = 0.
- Range: All non-negative real numbers, which can be expressed as [0, +∞).
In summary, for the function f(x) = x6:
- Domain: (-∞, +∞)
- Range: [0, +∞)