Function F(x) = √(-x): Which Statement Is True?

by Alex Johnson 48 views

Hey there, math enthusiasts! Ever stumbled upon a function that looks a bit tricky and makes you scratch your head? Today, we're diving deep into the function f(x) = √(-x). This one might seem simple at first glance, but it holds some interesting mathematical concepts within. We're going to dissect it piece by piece, and by the end, you'll not only understand it but also be able to explain it to others. So, let’s get started and unravel the mystery behind this square root function!

Understanding the Domain

When we talk about the domain of a function, we're essentially asking: “What are all the possible x-values that we can plug into the function and get a real number as an output?” For the function f(x) = √(-x), this is where things get interesting. Remember, we can't take the square root of a negative number and get a real number result. For example, √(-4) is not a real number; it's an imaginary number (2i). So, to avoid ending up in the realm of imaginary numbers, what do we need to ensure about the value inside the square root, which is -x in our case?

We need -x to be greater than or equal to zero. Mathematically, we write this as -x ≥ 0. Now, let's solve this inequality. If we multiply both sides by -1, we need to flip the inequality sign (a crucial step!). This gives us x ≤ 0. What does this mean? It means that the only x-values that we can use in this function are those that are less than or equal to zero. If we plug in a positive number, say x = 4, we get f(4) = √(-4), which is not a real number. But if we plug in x = -4, we get f(-4) = √(-(-4)) = √(4) = 2, which is a perfectly valid real number. Therefore, the domain of the function f(x) = √(-x) is all real numbers less than or equal to 0. This is a key point to grasp, as it dictates the very foundation of our function.

Exploring the Range

Now that we've conquered the domain, let's shift our focus to the range of the function. The range is all the possible output values (or y-values) that the function can produce. For f(x) = √(-x), we know that the square root function, by definition, always returns a non-negative value. Think about it: the square root of a number is the value that, when multiplied by itself, gives you the original number. So, the square root of 9 is 3 (because 3 * 3 = 9), and we only consider the positive root. This means that the output of √(-x) will always be greater than or equal to zero.

But how high can these output values go? Well, as x becomes a larger negative number, -x becomes a larger positive number, and the square root of that larger positive number also becomes larger. For instance, if x = -100, then f(-100) = √(100) = 10. If x = -10000, then f(-10000) = √(10000) = 100. We can see that the output values can grow indefinitely as x becomes increasingly negative. However, they will never be negative themselves. This tells us that the range of the function f(x) = √(-x) is all non-negative real numbers, or y ≥ 0. This understanding of the range is crucial for visualizing the behavior of the function.

Graphing the Function

To truly understand a function, it's incredibly helpful to visualize it by graphing it. When we graph f(x) = √(-x), we'll see a curve that starts at the point (0, 0) and extends to the left, in the second quadrant of the coordinate plane. Remember our discussion about the domain? It's all the x-values less than or equal to zero, which is why the graph exists only on the left side of the y-axis. And our range? It's all the y-values greater than or equal to zero, which is why the graph is only above the x-axis.

The graph starts at the origin (0, 0) because f(0) = √(-0) = √(0) = 0. As x becomes more negative, the graph rises. For example, at x = -1, f(-1) = √(1) = 1, so we have the point (-1, 1). At x = -4, f(-4) = √(4) = 2, giving us the point (-4, 2). If you were to plot these points and connect them, you'd see a smooth curve that increases as it moves to the left. This visual representation solidifies our understanding of both the domain and the range and provides a clear picture of the function's behavior. Graphing is a powerful tool in mathematics, and it's always worth taking the time to sketch out a function to gain a deeper insight.

The Correct Statement

Now that we've thoroughly explored the domain, range, and graph of f(x) = √(-x), let's revisit the original question: Which statement is true about this function?

  • A. The domain of the graph is all real numbers.
  • B. The range of the graph is all real numbers.
  • C. The domain of the graph is all real numbers less than or equal to 0.
  • D. The range of the graph is all real numbers less than or equal to 0.

Based on our analysis, we know that statement C is the correct one. The domain of f(x) = √(-x) is indeed all real numbers less than or equal to 0. Statement A is incorrect because, as we discussed, we can't plug in positive numbers for x. Statement B is incorrect because the range is all non-negative real numbers, not all real numbers. And statement D is incorrect because the range is greater than or equal to 0, not less than or equal to 0.

Real-World Applications and Further Exploration

You might be wondering, “Okay, this is interesting, but where would I ever use this in the real world?” While f(x) = √(-x) itself might not have direct, everyday applications, understanding domains, ranges, and how functions behave is crucial in many fields. For example, in physics, you might use functions to model the trajectory of a projectile, and the domain would represent the time interval during which the projectile is in motion. In economics, functions can model cost and revenue, and understanding the domain and range helps to make sense of the model's limitations and realistic outputs.

If you're eager to delve deeper, you could explore related topics like transformations of functions. What happens if we add a constant to f(x), like f(x) = √(-x) + 2? How does this shift the graph? What if we multiply x by a constant inside the square root, like f(x) = √(-2x)? These types of explorations can significantly enhance your understanding of functions and their behavior.

Conclusion

So, there you have it! We've dissected the function f(x) = √(-x), understood its domain and range, visualized its graph, and pinpointed the correct statement about it. We've seen how crucial it is to consider the limitations imposed by square roots and how these limitations shape the function's behavior. Remember, understanding the domain and range is fundamental to grasping the essence of any function.

Keep exploring, keep questioning, and keep enjoying the fascinating world of mathematics! For further learning and to solidify your understanding of functions, you can explore resources like Khan Academy's section on domain and range. Happy learning!