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.
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.
The parent fifth-root function is denoted as f(x) = √[5]{x}. Let's analyze its key characteristics:
1. Domain: The domain of the fifth-root function is all real numbers since the root can be taken of any real number.
2. Range: The range of the parent fifth-root function is also all real numbers since the root can produce both positive and negative values.
3. Intervals of increasing, decreasing, or constant: The fifth-root function is always increasing for positive values of x, decreasing for negative values of x, and constant at zero (f(0) = 0).
4. Intercepts: The x-intercept occurs when f(x) = 0, which happens at x = 0. There are no y-intercepts since the function starts at (0, 0) and extends infinitely.
5. Even, odd, or neither: The fifth-root function is neither even nor odd since f(-x) does not equal -f(x) or f(x).
6. Continuity: The parent fifth-root function is continuous over its entire domain. There are no breaks or jumps in the graph.
7. Asymptotes: There are no vertical asymptotes for the fifth-root function since the root can be taken for any real number. However, there is a horizontal asymptote at y = 0 as x approaches infinity or negative infinity.
8. End behavior: As x approaches positive or negative infinity, the graph of the fifth-root function approaches the x-axis, displaying a horizontal end behavior.
To summarize:
Domain: All real numbers
Range: All real numbers
Increasing intervals: Positive x-values
Decreasing intervals: Negative x-values
Constant: At x = 0
Intercepts: X-intercept at (0, 0)
Even/Odd: Neither
Continuity: Continuous
Asymptotes: Horizontal asymptote y = 0 (x approaches infinity)
End behavior: Approaches x-axis as x approaches infinity or negative infinity
To identify the key characteristics of the parent fifth-root function, let's break it down step by step:
1. Domain: The domain of the parent fifth-root function, also known as the radical function, is all real numbers because you can take the fifth root of any real number.
2. Range: The range of the fifth-root function is also all real numbers, as the output of taking the fifth root can be any real number.
3. Intervals of Increase, Decrease, and Constant: The fifth-root function is always increasing because as you move to the right on the number line, the value of the function also increases. It never decreases or remains constant.
4. Intercepts: The fifth-root function does not have any x-intercepts since taking the fifth root of any number (except 0) results in a nonzero value. However, it does have a y-intercept, which occurs at (0, 0) since the fifth root of 0 is 0.
5. Even or Odd Function: The parent fifth-root function is neither even nor odd. An even function has symmetry about the y-axis, which means that f(x) = f(-x) for all x in the domain. An odd function has symmetry about the origin, which means that f(x) = -f(-x) for all x in the domain. However, the fifth-root function doesn't possess these properties.
6. Continuity: The parent fifth-root function is continuous over its entire domain, as there are no breaks, jumps, or holes in the graph.
7. Asymptotes: The fifth-root function doesn't have any vertical or horizontal asymptotes. As x approaches positive or negative infinity, the function approaches positive or negative infinity, respectively. However, this doesn't constitute an asymptote.
8. End Behavior: As x approaches positive or negative infinity, the fifth-root function exhibits infinite growth. In other words, the function gets larger and larger, but it does not approach a specific value.
By considering these characteristics, you can analyze and understand the key features of the parent fifth-root function.