Identify the key characteristics of the parent fifth-root function . Include the following: domain, range, intervals where the function is increasing, decreasing, or constant, intercepts, whether the function is even, odd, or neither, whether the function is continuous or discontinuous, asymptotes, and end behavior.

Answer 1

The parent fifth-root function, denoted as f(x) = ∛x or f(x) = x^(1/5), has the following characteristics:

Domain:
The domain of the fifth-root function is all real numbers since we can take the fifth root of any real number.

Range:
The range of the fifth-root function is also all real numbers as the function can output any value when given a real number as input.

Intervals of Increase/Decrease/Constant:
The fifth-root function is always increasing. As x increases, the function's output also increases.

Intercepts:
The x-intercept of the fifth-root function is (0, 0) since ∛0 = 0. The function does not have a y-intercept since there is no real number that, when taken to the power of 1/5, equals 0.

Even, Odd, or Neither:
The fifth-root function is neither even nor odd since it does not satisfy the conditions for either type of symmetry. Specifically, it does not hold f(x) = f(-x) for even functions and f(-x) = -f(x) for odd functions.

Continuity:
The fifth-root function is continuous for all real numbers. There are no points where the function is discontinuous.

Asymptotes:
The fifth-root function does not have any asymptotes.

End Behavior:
As x approaches negative or positive infinity, f(x) approaches negative infinity or positive infinity, respectively. This means that the function has no defined end behavior.