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Exploring the Range and Graph of (f(x) |x| cdot x)
Exploring the Range and Graph of (f(x) |x| cdot x)
Let's delve into analyzing the function (f(x) |x| cdot x). Understanding the behavior, domain, and range of this function is crucial for gaining insightful knowledge in calculus and practical applications involving absolute values.
Domain
The domain of (f(x) |x| cdot x) is all real numbers (mathbb{R}) since the expression involves both (x^2) and (x), both of which are defined for all real numbers.
Behavior of the Function
To analyze (f(x) |x| cdot x), we consider two cases based on the definition of the absolute value function:
Case 1: (x geq 0)
In this case, (|x| x). Therefore, the function simplifies to:
[f(x) x cdot x x^2]This is a standard quadratic function that opens upwards and has its vertex at (x 0).
Case 2: (x
Here, (|x| -x). Hence, the function becomes:
[f(x) -x cdot x -x^2]This is also a quadratic function, opening downwards and similarly, its vertex is at (x 0).
Finding the Range
For (x geq 0): The function simplifies to (f(x) x^2). This is a quadratic function that opens upwards and has its vertex at (x 0), which is the minimum point. At (x 0), (f(0) 0^2 0). As (x) increases, (f(x)) increases without bound. Therefore, the range for (x geq 0) is ([0, infty)). For (x The function simplifies to (f(x) -x^2). This quadratic function also opens upwards, but the sign is inverted due to the negative (x^2) term. Since (x^2) always dominates the linear term, as (x) approaches (0) from the left (e.g., (x -0.1)), (f(-0.1) -(-0.1)^2 -0.01) approaches (0). As (x) decreases and becomes more negative (for example, (x -1)), (f(-1) -(-1)^2 -1). Consequently, the function approaches infinity as (x) becomes more negative. The minimum value occurs at (x 0) and is (0).Conclusion: The overall range of (f(x) |x| cdot x) is ([0, infty)) since the behavior in both cases is to increase from (0) onwards.
Graph of the Function
To graph (f(x) |x| cdot x), we consider both cases:
For (x geq 0):
The graph is a parabola opening upwards, with its vertex at ((0,0)).
For (x
The graph is also a parabola opening downwards, but this part of the graph is reflected over the y-axis due to the (|x|) term. As (x) becomes more negative, the function values approach (-infty).
A rough sketch of the function is provided below:
A rough sketch of the graph of (f(x) |x| cdot x). The graph shows a parabolic shape for both positive and negative values of (x), with the overall shape indicating an increasing function as (x) increases.The graph starts at ((0,0)) and increases in both directions, showing the parabolic shape for both positive and negative values of (x).
The minimum point is at ((0,0)), and the function approaches (infty) as (x) increases in the positive direction and approaches (-infty) as (x) decreases in the negative direction.