For nth root functions, I know that if "n" is even, then the domain must be equal or greater than zero. If "n" is odd, then domain will be all set of real numbers. But how do I determine the range of an n-th root function?
To determine the range of an nth root function, you need to consider two cases: when n is even and when n is odd.
1. When n is even:
- If n is even and the function has a positive coefficient (e.g., f(x) = ax^(1/n)), then the range will be equal to or greater than zero.
- If n is even and the function has a negative coefficient (e.g., f(x) = -ax^(1/n)), then the range will be less than or equal to zero. In this case, the values of the function will be negative or zero.
2. When n is odd:
- If n is odd, the range will be all real numbers. This means that the function can take any positive value, negative value, or zero.
In summary:
- When n is even and the coefficient is positive, the range is equal to or greater than zero.
- When n is even and the coefficient is negative, the range is less than or equal to zero.
- When n is odd, the range is all real numbers.
Remember to consider the coefficient when determining the range, as it can affect whether the values are positive or negative.
To determine the range of an n-th root function, you can follow a similar approach as you did for the domain.
1. Determine the possible values of the function.
- If "n" is even, the function will output real numbers, including zero, since the domain is non-negative.
- If "n" is odd, the function will output all real numbers, as there are no restrictions on the domain.
2. Determine the behavior of the function as the input approaches positive infinity and negative infinity.
- If "n" is even, as the input approaches positive infinity, the function output will approach zero. As the input approaches negative infinity, the function output will also approach zero.
- If "n" is odd, as the input approaches positive or negative infinity, the function output will approach positive or negative infinity, respectively.
3. Consider any other restrictions or limitations.
- If there are any specific restrictions or limitations stated in the problem or equation, such as a lower or upper bound on the domain, you need to incorporate them into determining the range.
By considering these factors, you can determine the range of an n-th root function.
the range is always (-∞,+∞) (if you allow both positive and negative roots of even powers)
just look at the graphs.