Analyzing The Function F(x) = √(-x): Domain And Range

by Alex Johnson 54 views

Let's dive into the function f(x) = √(-x) and figure out its domain and range. This is a classic problem in mathematics that helps us understand how functions behave, especially when dealing with square roots and negative signs. Understanding the domain and range is crucial for grasping the complete picture of a function's behavior. The domain tells us all the possible input values (x-values) that the function can accept, while the range tells us all the possible output values (y-values) that the function can produce. So, let's break it down step by step.

Understanding the Function f(x) = √(-x)

First, let's take a closer look at the function f(x) = √(-x). The key thing to notice here is the square root and the negative sign inside the square root. Remember, in the world of real numbers, we can only take the square root of non-negative numbers (i.e., 0 or positive numbers). This is because the square root of a negative number is not a real number; it's an imaginary number. This constraint significantly impacts the domain of our function. The negative sign inside the square root also plays a crucial role. It essentially flips the sign of our input x before we take the square root. This means that if we plug in a positive number for x, the expression inside the square root becomes negative, which is a no-go in the real number system. On the other hand, if we plug in a negative number for x, the expression inside the square root becomes positive, which is perfectly fine. Understanding this interplay between the square root and the negative sign is the first step in determining the function's domain. We also need to consider what happens when x is zero. If x is zero, then -x is also zero, and the square root of zero is zero, which is a valid real number. So, zero is definitely part of our function's domain. Keeping these points in mind, we can start narrowing down the possible values of x that make our function happy.

Determining the Domain

The domain of a function is the set of all possible input values (x-values) for which the function produces a real number output. In the case of f(x) = √(-x), we need to ensure that the expression inside the square root, which is -x, is greater than or equal to zero. Mathematically, we can write this as:

  • (-x) ≥ 0

To solve this inequality, we can multiply both sides by -1. Remember, when we multiply or divide an inequality by a negative number, we need to flip the inequality sign. So, we get:

  • x ≤ 0

This inequality tells us that x must be less than or equal to zero for the function to be defined in the real number system. In other words, the domain of f(x) consists of all real numbers that are zero or negative. We can visualize this on a number line as a ray extending from 0 to the left, including 0. Now, let's think about why positive numbers are not allowed in the domain. If we try to plug in a positive number for x, say x = 4, then we get:

  • f(4) = √(-4)

The square root of -4 is not a real number because there is no real number that, when multiplied by itself, gives -4. This is why positive numbers are excluded from the domain. On the other hand, if we plug in a negative number, say x = -9, then we get:

  • f(-9) = √(-(-9)) = √(9) = 3

This is a real number, so -9 is in the domain. Similarly, if we plug in x = 0, we get:

  • f(0) = √(-0) = √(0) = 0

So, 0 is also in the domain. Based on this analysis, we can confidently say that the domain of f(x) = √(-x) is all real numbers less than or equal to 0.

Determining the Range

The range of a function is the set of all possible output values (y-values) that the function can produce. To find the range of f(x) = √(-x), we need to consider what happens to the output as we vary the input x within its domain (i.e., x ≤ 0). Let's think about the square root function first. The square root function always returns a non-negative value. This is because the square root of a number is defined as the non-negative number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, not -3, even though both 3 and -3, when squared, give 9. This means that the output of the square root part of our function, √( -x ), will always be greater than or equal to zero. Now, let's consider the effect of the negative sign inside the square root. As we discussed earlier, the negative sign ensures that we are taking the square root of a non-negative number. As x varies from negative values to 0, -x varies from positive values to 0. The square root of a positive number is a positive number, and the square root of 0 is 0. So, as x varies from negative values to 0, the output of the square root function, √( -x ), varies from positive values to 0. This means that the range of f(x) includes 0 and all positive real numbers. There is no upper bound to the range because as x becomes more and more negative, -x becomes more and more positive, and the square root of -x also becomes more and more positive. For example, if x = -100, then:

  • f(-100) = √(-(-100)) = √(100) = 10

If x = -10000, then:

  • f(-10000) = √(-(-10000)) = √(10000) = 100

And so on. We can see that the output can become arbitrarily large as x becomes more and more negative. Therefore, the range of f(x) = √(-x) is all non-negative real numbers, which means all real numbers greater than or equal to 0. In mathematical notation, we can write this as y ≥ 0.

Analyzing the Given Statements

Now that we have a clear understanding of the domain and range of f(x) = √(-x), let's evaluate the statements given in the question:

  • A. The domain of the graph is all real numbers. This statement is incorrect. We determined that the domain is all real numbers less than or equal to 0.
  • B. The range of the graph is all real numbers. This statement is also incorrect. We found that the range is all real numbers greater than or equal to 0.
  • C. The domain of the graph is all real numbers less than or equal to 0. This statement is correct. This matches our analysis of the domain.
  • D. The range of the graph is all real numbers less than or equal to 0. This statement is incorrect. The range is all real numbers greater than or equal to 0.

Therefore, the correct statement is C. The domain of the graph is all real numbers less than or equal to 0.

Conclusion

In conclusion, by carefully analyzing the function f(x) = √(-x), we have determined that its domain is all real numbers less than or equal to 0, and its range is all real numbers greater than or equal to 0. This exercise highlights the importance of understanding the constraints imposed by mathematical operations like square roots and negative signs when determining the domain and range of a function. Remember, the domain and range give us a complete picture of how a function behaves, allowing us to make accurate predictions and solve related problems. For further exploration of functions, domains, and ranges, you might find helpful resources on websites like Khan Academy's Algebra Section.