If F ( X ) = X 2 + 9 F(x)=\sqrt{x^2+9} F ( X ) = X 2 + 9 ​ , Then F ˋ ( − 4 ) = \grave{f}(-4)= F ˋ ​ ( − 4 ) = (a) − 4 5 \frac{-4}{5} 5 − 4 ​ (b) 5 (c) 1 10 \frac{1}{10} 10 1 ​ (d) − 1 10 \frac{-1}{10} 10 − 1 ​

In mathematics, functions play a crucial role in representing relationships between variables. Given a function $f(x)=\sqrt$x^2+9}x2+9​, we are tasked with finding the value of $\grave{f(-4)$. To accomplish this, we need to substitute $-4$ into the function and simplify the expression.

The Function and Its Properties

The given function is $f(x)=\sqrt{x^2+9}$. This function represents a square root of the sum of $x^2$ and $9$. The square root function is defined only for non-negative real numbers, which means that the expression inside the square root must be non-negative.

Substituting $-4$ into the Function

To find the value of $\grave{f}(-4)$, we substitute $-4$ into the function in place of $x$. This gives us:

$$\grave{f}(-4)=\sqrt{(-4)^2+9}$$

Simplifying the Expression

Now, we simplify the expression inside the square root:

$$\grave{f}(-4)=\sqrt{16+9}$$

$$\grave{f}(-4)=\sqrt{25}$$

Evaluating the Square Root

The square root of $25$ is $5$, since $5^2=25$. Therefore, we have:

$$\grave{f}(-4)=5$$

Conclusion

In conclusion, the value of $\grave{f}(-4)$ is $5$. This is the correct answer among the given options.

Why is the Answer $5$?

The answer $5$ is obtained by substituting $-4$ into the function and simplifying the expression. The function $f(x)=\sqrt{x^2+9}$ represents a square root of the sum of $x^2$ and $9$. When we substitute $-4$ into the function, we get $\sqrt{(-4)^2+9}$. Simplifying this expression gives us $\sqrt{25}$, which is equal to $5$.

What is the Significance of the Function?

The function $f(x)=\sqrt{x^2+9}$ represents a square root of the sum of $x^2$ and $9$. This function is significant because it can be used to model real-world situations where the relationship between variables is represented by a square root. For example, the function can be used to model the relationship between the distance of an object from a fixed point and the time it takes to travel that distance.

What are the Applications of the Function?

The function $f(x)=\sqrt{x^2+9}$ has several applications in mathematics and other fields. Some of the applications of the function include:

• Modeling real-world situations: The function can be used to model real-world situations where the relationship between variables is represented by a square root.
• Solving equations: The function can be used to solve equations that involve square roots.
• Graphing functions: The function can be used to graph functions that involve square roots.

What are the Limitations of the Function?

The function $f(x)=\sqrt{x^2+9}$ has several limitations. Some of the limitations of the function include:

• Domain restriction: The function is defined only for non-negative real numbers, which means that the expression inside the square root must be non-negative.
• Range restriction: The function is defined only for non-negative real numbers, which means that the output of the function must be non-negative.

Conclusion

$$ $$

Q: What is the domain of the function $f(x)=\sqrt{x^2+9}$?

A: The domain of the function $f(x)=\sqrt{x^2+9}$ is all real numbers, since the expression inside the square root is always non-negative.

Q: What is the range of the function $f(x)=\sqrt{x^2+9}$?

A: The range of the function $f(x)=\sqrt{x^2+9}$ is all non-negative real numbers, since the output of the function is always non-negative.

Q: How do I evaluate the function $f(x)=\sqrt{x^2+9}$ at a given value of $x$?

A: To evaluate the function $f(x)=\sqrt{x^2+9}$ at a given value of $x$, substitute the value of $x$ into the function and simplify the expression.

Q: What is the value of $\grave{f}(-4)$?

A: The value of $\grave{f}(-4)$ is $5$, since $\grave{f}(-4)=\sqrt{(-4)^2+9}=\sqrt{25}=5$.

Q: Can I use the function $f(x)=\sqrt{x^2+9}$ to model real-world situations?

A: Yes, the function $f(x)=\sqrt{x^2+9}$ can be used to model real-world situations where the relationship between variables is represented by a square root.

Q: What are some applications of the function $f(x)=\sqrt{x^2+9}$?

A: Some applications of the function $f(x)=\sqrt{x^2+9}$ include:

• Modeling real-world situations: The function can be used to model real-world situations where the relationship between variables is represented by a square root.
• Solving equations: The function can be used to solve equations that involve square roots.
• Graphing functions: The function can be used to graph functions that involve square roots.

Q: What are some limitations of the function $f(x)=\sqrt{x^2+9}$?

A: Some limitations of the function $f(x)=\sqrt{x^2+9}$ include:

• Domain restriction: The function is defined only for non-negative real numbers, which means that the expression inside the square root must be non-negative.
• Range restriction: The function is defined only for non-negative real numbers, which means that the output of the function must be non-negative.

Q: Can I use the function $f(x)=\sqrt{x^2+9}$ to solve equations that involve square roots?

A: Yes, the function $f(x)=\sqrt{x^2+9}$ can be used to solve equations that involve square roots.

Q: How do I graph the function $f(x)=\sqrt{x^2+9}$?

A: To graph the function $f(x)=\sqrt{x^2+9}$, use a graphing calculator or software to plot the function.

Q: What is the significance of the function $f(x)=\sqrt{x^2+9}$?

A: The function $f(x)=\sqrt{x^2+9}$ is significant because it can be used to model real-world situations where the relationship between variables is represented by a square root.

Conclusion

In conclusion, the function $f(x)=\sqrt{x^2+9}$ is a significant function that has several applications in mathematics and other fields. The function can be used to model real-world situations, solve equations, and graph functions that involve square roots. However, the function has several limitations, including domain and range restrictions.