Points On The Graph Of F(x) = -√(-x) A Step-by-Step Solution
In the realm of mathematics, understanding the behavior of functions and their graphical representations is crucial. This article delves into the intricacies of the function f(x) = -√(-x), aiming to identify which points lie on its graph. We will embark on a journey to explore the domain, range, and key characteristics of this function, ultimately determining which of the provided points satisfy its equation. Whether you're a student grappling with function analysis or a math enthusiast seeking a deeper understanding, this guide will equip you with the knowledge to confidently navigate the world of graphs and functions.
Understanding the Function f(x) = -√(-x)
Before we dive into specific points, let's first dissect the function f(x) = -√(-x) itself. The key to understanding this function lies in recognizing its components: the square root and the negative signs. The square root function, denoted by √, only accepts non-negative inputs. This means that the expression inside the square root, in this case, -x, must be greater than or equal to zero. Mathematically, we express this as -x ≥ 0. Multiplying both sides by -1 (and flipping the inequality sign) gives us x ≤ 0. This tells us that the domain of the function, the set of all possible input values (x-values), is all real numbers less than or equal to zero. In interval notation, the domain is (-∞, 0].
Next, let's consider the impact of the negative signs. The negative sign inside the square root, -x, ensures that we're taking the square root of a non-negative number when x is negative. The negative sign outside the square root, -√(-x), then reflects the output of the square root function across the x-axis. Since the square root function always produces non-negative outputs, the negative sign in front makes the entire function's output non-positive. This means the range of the function, the set of all possible output values (y-values), is all real numbers less than or equal to zero. In interval notation, the range is (-∞, 0].
The function f(x) = -√(-x), therefore, represents a square root function that is reflected across both the x-axis (due to the negative sign outside the square root) and the y-axis (due to the negative sign inside the square root). This reflection results in a graph that starts at the origin (0, 0) and extends into the second quadrant (where x is negative and y is negative). This foundational understanding is crucial for accurately determining which points lie on the graph.
H2: Testing Points Against the Function
Now that we have a solid grasp of the function's behavior, let's examine the provided points and determine whether they satisfy the equation f(x) = -√(-x). To do this, we will substitute the x-coordinate of each point into the function and see if the resulting y-value matches the point's y-coordinate. This process involves careful evaluation of the square root and attention to the signs. Each point will be analyzed individually to ensure accuracy and clarity.
Point 1: (-9, 3)
For the point (-9, 3), we substitute x = -9 into the function:
f(-9) = -√(-(-9)) = -√(9) = -3
Since f(-9) = -3, and the y-coordinate of the point is 3, this point does not lie on the graph of the function. The function yields a y-value of -3 when x is -9, but the point has a y-value of 3. This discrepancy indicates that the point does not satisfy the function's equation.
Point 2: (-4, -2)
Next, let's consider the point (-4, -2). Substituting x = -4 into the function, we get:
f(-4) = -√(-(-4)) = -√(4) = -2
In this case, f(-4) = -2, which matches the y-coordinate of the point. Therefore, the point (-4, -2) lies on the graph of f(x) = -√(-x). This point satisfies the function's equation, confirming its presence on the graph.
Point 3: (-1, 1)
Now, let's evaluate the point (-1, 1). Substituting x = -1 into the function:
f(-1) = -√(-(-1)) = -√(1) = -1
Since f(-1) = -1, and the y-coordinate of the point is 1, this point does not lie on the graph of the function. The function's output is -1 when x is -1, but the point has a y-value of 1, indicating it does not satisfy the equation.
Point 4: (0, 0)
For the point (0, 0), we substitute x = 0 into the function:
f(0) = -√(-(0)) = -√(0) = 0
Here, f(0) = 0, which matches the y-coordinate of the point. Thus, the point (0, 0) lies on the graph of f(x) = -√(-x). This point satisfies the function's equation, placing it on the graph.
Point 5: (1, 1)
Let's analyze the point (1, 1). Substituting x = 1 into the function:
f(1) = -√(-(1)) = -√(-1)
We immediately encounter a problem: we are trying to take the square root of a negative number. As we established earlier, the square root function only accepts non-negative inputs. Therefore, f(1) is undefined. Consequently, the point (1, 1) does not lie on the graph of the function. Furthermore, since the domain of the function is x ≤ 0, any point with a positive x-coordinate cannot be on the graph.
Point 6: (2, -4)
For the point (2, -4), we substitute x = 2 into the function:
f(2) = -√(-(2)) = -√(-2)
Similar to the previous point, we are attempting to take the square root of a negative number. This means that f(2) is undefined. Therefore, the point (2, -4) does not lie on the graph of the function. Again, the positive x-coordinate violates the domain of the function.
Point 7: (9, 3)
Finally, let's examine the point (9, 3). Substituting x = 9 into the function:
f(9) = -√(-(9)) = -√(-9)
Once again, we encounter the square root of a negative number, making f(9) undefined. Consequently, the point (9, 3) does not lie on the graph of the function. The positive x-coordinate is outside the function's domain.
H2: Summary of Points on the Graph
After meticulously evaluating each point, we have determined the following:
- (-9, 3): Does not lie on the graph.
- (-4, -2): Lies on the graph.
- (-1, 1): Does not lie on the graph.
- (0, 0): Lies on the graph.
- (1, 1): Does not lie on the graph.
- (2, -4): Does not lie on the graph.
- (9, 3): Does not lie on the graph.
Therefore, only the points (-4, -2) and (0, 0) lie on the graph of the function f(x) = -√(-x). This conclusion is based on our understanding of the function's domain, range, and the substitution of point coordinates into the function's equation.
H2: Graphing the Function
To further solidify our understanding, let's visualize the graph of f(x) = -√(-x). As we discussed earlier, the graph starts at the origin (0, 0) and extends into the second quadrant. The points we identified as lying on the graph, (-4, -2) and (0, 0), will be present on the curve. The graph will have a shape similar to a square root function, but reflected across both the x-axis and the y-axis.
[It would be beneficial to include an actual graph image here, but since I am a text-based AI, I cannot generate images. You can easily plot the graph using online graphing calculators or software like Desmos or Wolfram Alpha.]
By visualizing the graph, we can confirm that the points (-4, -2) and (0, 0) indeed lie on the curve, while the other points are located elsewhere in the coordinate plane. This visual representation serves as a powerful tool for reinforcing our analytical findings.
H2: Key Takeaways and Implications
This exploration of the function f(x) = -√(-x) has yielded several key takeaways:
- Domain and Range are Crucial: Understanding the domain and range of a function is essential for determining which points can potentially lie on its graph. In this case, the domain restriction (x ≤ 0) immediately eliminated several points from consideration.
- Substitution is Key: The primary method for verifying whether a point lies on a graph is to substitute its x-coordinate into the function and compare the result to the point's y-coordinate.
- Visual Representation Enhances Understanding: Graphing the function provides a visual confirmation of our analytical results and helps to solidify our understanding of the function's behavior.
The ability to analyze functions and their graphs is a fundamental skill in mathematics. This exercise demonstrates how to systematically approach such problems, combining analytical techniques with visual aids to arrive at accurate conclusions. The implications of this understanding extend to various fields, including physics, engineering, and computer science, where mathematical functions are used to model real-world phenomena.
H2: Conclusion
In conclusion, we have successfully identified the points that lie on the graph of the function f(x) = -√(-x). Through a combination of function analysis, point substitution, and graphical visualization, we determined that only the points (-4, -2) and (0, 0) satisfy the function's equation and therefore reside on its graph. This comprehensive exploration highlights the importance of understanding function properties, applying analytical techniques, and utilizing visual aids in mathematical problem-solving. By mastering these skills, we can confidently navigate the world of functions and graphs, unlocking their potential to model and explain the world around us.
