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.
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) → +∞.
To identify the key characteristics of the parent fifth root function f(x) = 5√x, let's examine each aspect:
1. Domain: The domain of the function consists of all non-negative real numbers, as the fifth root can only be taken of non-negative values (x ≥ 0).
2. Range: The range of the function includes all non-negative real numbers, as the fifth root of any non-negative number is also non-negative (f(x) ≥ 0).
3. Intervals of increasing, decreasing, or constant behavior: The function is always increasing within its domain (x ≥ 0) since as x increases, the fifth root of x also increases. There are no intervals of decreasing or constant behavior.
4. Intercepts:
- x-intercept: The x-intercept occurs when f(x) = 0. Since 5√x = 0, the only x that satisfies this equation is x = 0. Therefore, the x-intercept is (0, 0).
- y-intercept: The y-intercept occurs when x = 0. Substituting x = 0 into the equation gives f(0) = 5√0 = 0. Hence, the y-intercept is (0, 0).
5. Even or odd: The fifth root function f(x) = 5√x is neither even nor odd because it is neither symmetric with respect to the y-axis (even function) nor the origin (odd function).
6. Continuity: The parent fifth root function f(x) = 5√x is continuous for all x ≥ 0. There are no discontinuities.
7. Asymptotes: The function f(x) = 5√x does not have any vertical or horizontal asymptotes.
8. End behavior: As x approaches positive infinity (x → ∞), the function f(x) increases without bound.
To identify the key characteristics of the parent fifth root function f(x) = 5√x, we can analyze each aspect:
1. Domain: The domain of this function is all non-negative real numbers since taking the root of a negative number is not defined. Therefore, the domain is [0, ∞).
2. Range: The range is the set of all non-negative real numbers, as the fifth root of any positive number results in a non-negative value. Therefore, the range is [0, ∞).
3. Intervals of Increase, Decrease, and Constant: As this is a monotonically increasing function, it is always increasing on its entire domain [0, ∞). There are no intervals of decrease or constancy.
4. Intercepts: To find the x-intercept, we set f(x) = 0 and solve for x: 5√x = 0. Since the fifth root of any non-negative number is always non-negative, there are no x-intercepts. To find the y-intercept, we substitute x = 0 into the function: f(0) = 5√0 = 0. Hence, the y-intercept is (0, 0).
5. Even, Odd, or Neither: To determine if the function is even or odd, we evaluate f(-x) and compare it with f(x). Substituting -x into the function gives us f(-x) = 5√(-x). However, taking the fifth root of a negative number is not defined in the real number system. Therefore, the function is neither even nor odd.
6. Continuity: The parent fifth root function f(x) = 5√x is continuous on its entire domain, as there are no breaks or removable discontinuities.
7. Asymptotes: The fifth root function does not have any vertical or horizontal asymptotes. It approaches the x-axis as x approaches positive infinity, but does not touch or cross it.
8. End Behavior: As x approaches positive infinity (x → ∞), the function f(x) also approaches positive infinity (f(x) → ∞). Similarly, as x approaches negative infinity (x → -∞), the function does not exist since the fifth root is not defined for negative numbers.
In summary, the key characteristics of the parent fifth root function f(x) = 5√x are:
- Domain: [0, ∞)
- Range: [0, ∞)
- Intervals of Increase: [0, ∞)
- Intervals of Decrease: None
- Intervals of Constant: None
- Intercepts: x-intercept = None; y-intercept = (0, 0)
- Even, Odd, or Neither: Neither
- Continuity: Continuous on its entire domain
- Asymptotes: None
- End Behavior: As x → ∞, f(x) → ∞; as x → -∞, f(x) is undefined