Question

Identify the key characteristics of the parent fifth-root function f left parenthesis x right parenthesis equals root index 5 start root x end root. 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 is given by f(x) = √[5]√[x]. Here are its key characteristics:

Domain: The domain of the function is all real numbers greater than or equal to 0. (x ≥ 0)

Range: The range of the function is all real numbers. (-∞ < f(x) < ∞)

Intervals of Increase: The function is strictly increasing for all positive real numbers. (x > 0)

Intervals of Decrease: The function is strictly decreasing for all negative real numbers. (x < 0)

Intervals of Constant: The function is constant at x = 0.

Intercepts: The x-intercept occurs at (0, 0), and there is no y-intercept since the function does not intersect the y-axis.

Even or Odd: Since f(-x) = -(f(x)), the function is odd.

Continuity: The fifth-root function is continuous for all real numbers.

Asymptotes: There are no asymptotes for this function.

End Behavior: As x approaches positive or negative infinity, f(x) also approaches positive or negative infinity, respectively.