Question

Identify the key characteristics of the parent fifth root function f(x)= 5sqrtx. Include the following: domain, range, intervals where the function is increasing, decreasing, or constant, intercepts, whether the function is even, odd, or neither, wheter the function is continous or discontinous, asymptotes, and end behavior.

Answer 1

The key characteristics of the parent fifth root function f(x) = 5√x are as follows:

Domain: The domain of the function is all non-negative real numbers (x ≥ 0).

Range: The range of the function is all non-negative real numbers (f(x) ≥ 0).

Intervals of Increase/Decrease: The function increases for all positive real numbers (x > 0) and remains constant at f(x) = 0 for x = 0.

Intercepts: The function does not have a y-intercept since f(x) ≠ 0 when x = 0. However, it has an x-intercept at (0,0).

Even/Odd: The function is neither even nor odd since f(-x) = -5√x ≠ 5√x.

Continuity: The function is continuous for all values of x in its domain.

Asymptotes: The function does not have any asymptotes.

End Behavior: As x approaches positive infinity, f(x) approaches positive infinity. As x approaches negative infinity, f(x) approaches positive infinity as well, since negative numbers raised to the fifth root will yield positive values.

Summary:
- Domain: x ≥ 0
- Range: f(x) ≥ 0
- Intervals of Increase/Decrease: Increasing for x > 0, f(x) = 0 for x = 0
- Intercepts: x-intercept at (0,0)
- Even/Odd: Neither
- Continuity: Continuous for all values of x in its domain
- Asymptotes: None
- End Behavior: As x → ±∞, f(x) → +∞.