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.
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.
The parent fifth-root function is given by f(x) = √[5]{x}. Let's analyze its key characteristics:
1. Domain: The domain of the parent fifth-root function is all real numbers greater than or equal to zero. This is because the fifth root (√[5]{}) can be taken for any non-negative real value.
2. Range: The range of the parent fifth-root function is also all real numbers greater than or equal to zero. This is because the fifth root (√[5]{}) always gives a non-negative result.
3. Intervals of Increase/Decrease/Constant: The function is always increasing on its entire domain. As x increases, the fifth-root (√[5]{}) of x increases in value.
4. Intercepts: The x-intercept of the parent fifth-root function is (0, 0) since f(0) = √[5]{0} = 0. However, there are no y-intercepts as the function never takes negative values.
5. Even/Odd/Neither: The parent fifth-root function is neither even nor odd. This can be demonstrated by checking whether the function satisfies f(-x) = f(x) (even) or f(-x) = -f(x) (odd). In this case, neither of these conditions is met.
6. Continuity: The parent fifth-root function is continuous for all real numbers, including zero. There are no jumps or breaks in the graph.
7. Asymptotes: The parent fifth-root function does not have any asymptotes. It does not approach any specific value as x approaches infinity or negative infinity.
8. End Behavior: As x approaches positive infinity or negative infinity, the fifth-root function approaches positive infinity. This means that the function increases without bound as x goes to infinity.
In summary, the key characteristics of the parent fifth-root function are:
- Domain: All real numbers greater than or equal to zero.
- Range: All real numbers greater than or equal to zero.
- Increases throughout its domain.
- No intercepts other than the x-intercept at (0, 0).
- Neither even nor odd.
- Continuous for all real numbers.
- No asymptotes.
- End behavior: Increases without bound as x approaches positive or negative infinity.
To identify the key characteristics of the parent fifth-root function ƒ(x) = √[5]x, let's break it down:
1. Domain: The domain of the fifth-root function is all real numbers, since you can take the fifth root of any real number.
2. Range: The range of the fifth-root function is also all real numbers, since the outcome of the fifth root can be any real number.
3. Increasing, decreasing, or constant intervals: The fifth-root function is always increasing because as the input value increases, the fifth root of that value will also increase. There are no constant or decreasing intervals.
4. Intercepts: The x-intercept is where the function crosses the x-axis, meaning where f(x) = 0. For the fifth-root function, the x-intercept occurs when x = 0. The y-intercept is where the function crosses the y-axis, which is when x = 0 as well. Therefore, the function has only one intercept at the origin (0, 0).
5. Even, odd, or neither: The fifth-root function is neither even nor odd because it does not satisfy the criteria of even or odd functions. An even function satisfies f(x) = f(-x) for every x, while an odd function satisfies f(x) = -f(-x) for every x. However, neither of these conditions holds for the fifth-root function.
6. Continuity: The fifth-root function is continuous for all real values of x since there are no breaks or jumps in the function. It is a smooth curve without any discontinuities.
7. Asymptotes: The fifth-root function does not have any asymptotes. Asymptotes occur when a function approaches a certain value but never reaches it, but in the case of the fifth-root function, it continues indefinitely without approaching any specific value.
8. End behavior: The end behavior of the fifth-root function is determined by the values of x as they approach positive or negative infinity. For positive values of x approaching infinity, the function grows larger and larger since the fifth root of a positive number is positive. For negative values of x approaching negative infinity, the function also grows larger and larger but remains negative, as the fifth root of a negative number is negative.