The Nature of the Function

Oct 13, 2025 by windailys.com 80 views

The given function $f(x)=-\sqrt{-x}$ involves a square root and a negative sign, which can be a bit tricky to understand. In this article, we will delve into the graph of this function and determine which points lie on it.

The Nature of the Function

The function $f(x)=-\sqrt{-x}$ is a combination of two operations: taking the square root of a negative number and multiplying it by -1. To understand the behavior of this function, let's break it down into smaller parts.

The square root of a negative number is an imaginary number, denoted by $i$ . For example, $\sqrt{-1}=i$ .
When we multiply an imaginary number by -1, we get a real number. For instance, $-i$ is a real number.

Analyzing the Graph

To analyze the graph of $f(x)=-\sqrt{-x}$ , we need to consider the domain and range of the function.

The domain of the function is all real numbers less than or equal to 0, i.e., $x \leq 0$ .
The range of the function is all real numbers greater than or equal to 0, i.e., $f(x) \geq 0$ .

Now, let's consider the points given in the options and determine which ones lie on the graph of $f(x)=-\sqrt{-x}$ .

Option A: $(-9, 3)$

To check if this point lies on the graph, we need to substitute $x=-9$ into the function and see if we get $f(-9)=3$ .

\begin{align*} f(-9) & = -\sqrt{-(-9)} \ & = -\sqrt{9} \ & = -3 \end{align*}

Since $f(-9)=-3$ and not $3$ , this point does not lie on the graph.

Option B: $(-4, -2)$

Let's substitute $x=-4$ into the function and see if we get $f(-4)=-2$ .

\begin{align*} f(-4) & = -\sqrt{-(-4)} \ & = -\sqrt{4} \ & = -2 \end{align*}

This point lies on the graph.

Option C: $(-1, 1)$

To check if this point lies on the graph, we need to substitute $x=-1$ into the function and see if we get $f(-1)=1$ .

\begin{align*} f(-1) & = -\sqrt{-(-1)} \ & = -\sqrt{1} \ & = -1 \end{align*}

Since $f(-1)=-1$ and not $1$ , this point does not lie on the graph.

Option D: $(0, 0)$

Let's substitute $x=0$ into the function and see if we get $f(0)=0$ .

\begin{align*} f(0) & = -\sqrt{-0} \ & = -\sqrt{0} \ & = 0 \end{align*}

This point lies on the graph.

Option E: $(1, 1)$

To check if this point lies on the graph, we need to substitute $x=1$ into the function and see if we get $f(1)=1$ .

\begin{align*} f(1) & = -\sqrt{-1} \ & = -i \end{align*}

Since $f(1)=-i$ and not $1$ , this point does not lie on the graph.

Option F: $(2, -4)$

Let's substitute $x=2$ into the function and see if we get $f(2)=-4$ .

\begin{align*} f(2) & = -\sqrt{-2} \ & = -\sqrt{2}i \end{align*}

Since $f(2)=-\sqrt{2}i$ and not $-4$ , this point does not lie on the graph.

Option G: $(9, 3)$

To check if this point lies on the graph, we need to substitute $x=9$ into the function and see if we get $f(9)=3$ .

\begin{align*} f(9) & = -\sqrt{-9} \ & = -3i \end{align*}

Since $f(9)=-3i$ and not $3$ , this point does not lie on the graph.

Conclusion

In conclusion, the points that lie on the graph of $f(x)=-\sqrt{-x}$ are:

$(-4, -2)$
$(0, 0)$

$These points satisfy the function $f(x)=-\sqrt{-x}$ and lie on the graph. The other options do not satisfy the function and therefore do not lie on the graph.<br/> **Frequently Asked Questions (FAQs) about the Graph of $f(x)=-\sqrt{-x}$**$

In the previous article, we analyzed the graph of the function $f(x)=-\sqrt{-x}$ and determined which points lie on it. In this article, we will answer some frequently asked questions (FAQs) about the graph of this function.

Q: What is the domain of the function $f(x)=-\sqrt{-x}$ ?

A: The domain of the function $f(x)=-\sqrt{-x}$ is all real numbers less than or equal to 0, i.e., $x \leq 0$ .

Q: What is the range of the function $f(x)=-\sqrt{-x}$ ?

A: The range of the function $f(x)=-\sqrt{-x}$ is all real numbers greater than or equal to 0, i.e., $f(x) \geq 0$ .

Q: Why does the function $f(x)=-\sqrt{-x}$ involve a square root of a negative number?

A: The function $f(x)=-\sqrt{-x}$ involves a square root of a negative number because the square root of a negative number is an imaginary number, denoted by $i$ . When we multiply an imaginary number by -1, we get a real number.

Q: What is the significance of the negative sign in the function $f(x)=-\sqrt{-x}$ ?

A: The negative sign in the function $f(x)=-\sqrt{-x}$ is used to multiply the square root of a negative number by -1, resulting in a real number.

Q: Can we simplify the function $f(x)=-\sqrt{-x}$ ?

A: Yes, we can simplify the function $f(x)=-\sqrt{-x}$ by using the property of square roots that $\sqrt{-x} = i\sqrt{x}$ .

\begin{align*} f(x) & = -\sqrt{-x} \ & = -i\sqrt{x} \end{align*}

Q: How can we graph the function $f(x)=-\sqrt{-x}$ ?

A: To graph the function $f(x)=-\sqrt{-x}$ , we can use the following steps:

Determine the domain and range of the function.
Plot the points that lie on the graph, such as $(-4, -2)$ and $(0, 0)$ .
Draw a smooth curve through the plotted points to obtain the graph of the function.

Q: What are some real-world applications of the function $f(x)=-\sqrt{-x}$ ?

A: The function $f(x)=-\sqrt{-x}$ has several real-world applications, including:

Electrical engineering: The function is used to model the behavior of electrical circuits with negative resistance.
Signal processing: The function is used to filter out noise from signals.
Image processing: The function is used to enhance the contrast of images.

Conclusion

In conclusion, the graph of the function $f(x)=-\sqrt{-x}$ is a complex function that involves a square root of a negative number. The domain and range of the function are all real numbers less than or equal to 0 and all real numbers greater than or equal to 0, respectively. The function has several real-world applications, including electrical engineering, signal processing, and image processing.