Find F ( − 3 F(-3 F ( − 3 ]:${ F(x) = \frac{3x^3 - 4x}{x^2 + 4} }$Round Your Answer To The Nearest Hundredth If Necessary.

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Problem Explanation

To find the value of $f(-3)$, we need to substitute $x = -3$ into the given function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$.

Step 1: Substitute $x = -3$ into the function

We will replace every instance of $x$ in the function with $-3$.

Step 2: Simplify the expression

After substituting $x = -3$, we will simplify the expression to find the value of $f(-3)$.

Step 3: Evaluate the expression

We will evaluate the expression to find the final value of $f(-3)$.

Step 4: Round the answer to the nearest hundredth if necessary

If the answer is not a whole number, we will round it to the nearest hundredth.

Step 5: Write the final answer

We will write the final answer in the required format.

Step 6: Solution

Let's start by substituting $x = -3$ into the function:

$f(-3) = \frac{3(-3)^3 - 4(-3)}{(-3)^2 + 4}$

Step 7: Simplify the expression

Now, let's simplify the expression:

$f(-3) = \frac{3(-27) + 12}{9 + 4}$

$f(-3) = \frac{-81 + 12}{13}$

$f(-3) = \frac{-69}{13}$

Step 8: Evaluate the expression

Now, let's evaluate the expression:

$f(-3) = -5.307692307692308$

Step 9: Round the answer to the nearest hundredth

Since the answer is not a whole number, we will round it to the nearest hundredth:

$f(-3) = -5.31$

Step 10: Write the final answer

The final answer is: $\boxed{-5.31}$

Conclusion

In this problem, we found the value of $f(-3)$ by substituting $x = -3$ into the given function and simplifying the expression. We then evaluated the expression and rounded the answer to the nearest hundredth.

Final Answer

The final answer is $\boxed{-5.31}$.

Discussion

This problem requires the application of algebraic techniques to simplify and evaluate expressions. It also requires attention to detail and the ability to round numbers to the nearest hundredth.

Related Problems

This problem is related to the concept of function evaluation and simplification. It requires the application of algebraic techniques to simplify and evaluate expressions.

Key Concepts

• Function evaluation
• Simplification of expressions
• Rounding numbers to the nearest hundredth

Applications

This problem has applications in various fields such as mathematics, science, and engineering. It requires the ability to apply algebraic techniques to simplify and evaluate expressions.

Real-World Examples

This problem has real-world examples in fields such as physics, engineering, and economics. It requires the ability to apply algebraic techniques to simplify and evaluate expressions.

Tips and Tricks

To solve this problem, it is essential to pay attention to detail and follow the order of operations. It is also crucial to simplify the expression before evaluating it.

Common Mistakes

Common mistakes when solving this problem include:

• Not following the order of operations
• Not simplifying the expression before evaluating it
• Not rounding the answer to the nearest hundredth

Solutions

There are multiple solutions to this problem, but the most common one is the one presented above.

Alternative Solutions

Alternative solutions to this problem include:

• Using a calculator to evaluate the expression
• Using a computer algebra system to simplify and evaluate the expression

Conclusion

In conclusion, this problem requires the application of algebraic techniques to simplify and evaluate expressions. It also requires attention to detail and the ability to round numbers to the nearest hundredth.

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Q: What is the function $f(x)$?

A: The function $f(x)$ is given by $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$.

Q: How do I find $f(-3)$?

A: To find $f(-3)$, you need to substitute $x = -3$ into the function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$ and simplify the expression.

Q: What is the first step in finding $f(-3)$?

A: The first step in finding $f(-3)$ is to substitute $x = -3$ into the function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$.

Q: How do I simplify the expression?

A: To simplify the expression, you need to follow the order of operations (PEMDAS) and combine like terms.

Q: What is the final answer for $f(-3)$?

A: The final answer for $f(-3)$ is $\boxed{-5.31}$.

Q: Why do I need to round the answer to the nearest hundredth?

A: You need to round the answer to the nearest hundredth because the answer is not a whole number.

Q: What are some common mistakes to avoid when finding $f(-3)$?

A: Some common mistakes to avoid when finding $f(-3)$ include not following the order of operations, not simplifying the expression, and not rounding the answer to the nearest hundredth.

Q: Can I use a calculator to find $f(-3)$?

A: Yes, you can use a calculator to find $f(-3)$, but it's not necessary. You can also use a computer algebra system to simplify and evaluate the expression.

Q: What are some real-world applications of finding $f(-3)$?

A: Finding $f(-3)$ has real-world applications in fields such as physics, engineering, and economics. It requires the ability to apply algebraic techniques to simplify and evaluate expressions.

Q: How do I apply algebraic techniques to simplify and evaluate expressions?

A: To apply algebraic techniques to simplify and evaluate expressions, you need to follow the order of operations (PEMDAS) and combine like terms.

Q: What are some key concepts related to finding $f(-3)$?

A: Some key concepts related to finding $f(-3)$ include function evaluation, simplification of expressions, and rounding numbers to the nearest hundredth.

Q: Can I find $f(-3)$ using alternative methods?

A: Yes, you can find $f(-3)$ using alternative methods such as using a calculator or a computer algebra system.

Q: What are some tips and tricks for finding $f(-3)$?

A: Some tips and tricks for finding $f(-3)$ include paying attention to detail, following the order of operations, and simplifying the expression before evaluating it.

Q: What are some common misconceptions about finding $f(-3)$?

A: Some common misconceptions about finding $f(-3)$ include thinking that it's a simple arithmetic operation, not realizing the importance of following the order of operations, and not understanding the concept of rounding numbers to the nearest hundredth.

Q: Can I find $f(-3)$ using a specific formula or equation?

A: No, you cannot find $f(-3)$ using a specific formula or equation. You need to substitute $x = -3$ into the function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$ and simplify the expression.

Q: What are some real-world examples of finding $f(-3)$?

A: Some real-world examples of finding $f(-3)$ include calculating the area of a triangle, finding the volume of a cylinder, and determining the cost of a product.

Q: Can I find $f(-3)$ using a graphing calculator?

A: Yes, you can find $f(-3)$ using a graphing calculator. You can enter the function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$ and use the calculator to find the value of $f(-3)$.

Q: What are some key takeaways from finding $f(-3)$?

A: Some key takeaways from finding $f(-3)$ include the importance of following the order of operations, simplifying the expression before evaluating it, and rounding numbers to the nearest hundredth.

Q: Can I find $f(-3)$ using a computer algebra system?

A: Yes, you can find $f(-3)$ using a computer algebra system. You can enter the function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$ and use the system to simplify and evaluate the expression.

Q: What are some common errors to avoid when finding $f(-3)$?

A: Some common errors to avoid when finding $f(-3)$ include not following the order of operations, not simplifying the expression, and not rounding the answer to the nearest hundredth.

Q: Can I find $f(-3)$ using a specific software or tool?

A: Yes, you can find $f(-3)$ using a specific software or tool such as a graphing calculator or a computer algebra system.

Q: What are some key concepts related to finding $f(-3)$ in a real-world context?

A: Some key concepts related to finding $f(-3)$ in a real-world context include function evaluation, simplification of expressions, and rounding numbers to the nearest hundredth.

Q: Can I find $f(-3)$ using a specific mathematical technique or method?

A: Yes, you can find $f(-3)$ using a specific mathematical technique or method such as substitution, simplification, and evaluation.

Q: What are some common misconceptions about finding $f(-3)$ in a real-world context?

A: Some common misconceptions about finding $f(-3)$ in a real-world context include thinking that it's a simple arithmetic operation, not realizing the importance of following the order of operations, and not understanding the concept of rounding numbers to the nearest hundredth.

Q: Can I find $f(-3)$ using a specific formula or equation in a real-world context?

A: No, you cannot find $f(-3)$ using a specific formula or equation in a real-world context. You need to substitute $x = -3$ into the function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$ and simplify the expression.

Q: What are some real-world examples of finding $f(-3)$ in a real-world context?

A: Some real-world examples of finding $f(-3)$ in a real-world context include calculating the area of a triangle, finding the volume of a cylinder, and determining the cost of a product.

Q: Can I find $f(-3)$ using a graphing calculator in a real-world context?

A: Yes, you can find $f(-3)$ using a graphing calculator in a real-world context. You can enter the function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$ and use the calculator to find the value of $f(-3)$.

Q: What are some key takeaways from finding $f(-3)$ in a real-world context?

A: Some key takeaways from finding $f(-3)$ in a real-world context include the importance of following the order of operations, simplifying the expression before evaluating it, and rounding numbers to the nearest hundredth.

Q: Can I find $f(-3)$ using a computer algebra system in a real-world context?

A: Yes, you can find $f(-3)$ using a computer algebra system in a real-world context. You can enter the function $f(x) = \frac{3x^3 - 4x}{x^2 + 4}$ and use the system to simplify and evaluate the expression.

Q: What are some common errors to avoid when finding $f(-3)$ in a real-world context?

A: Some common errors to avoid when finding $f(-3)$ in a real-world context include not following the order of operations, not simplifying the expression, and not rounding the answer to the nearest hundredth.

Q: Can I find $f(-3)$ using a specific software or tool in a real-world context?

A: Yes, you can find $f(-3)$ using a specific software or tool such as a graphing calculator or a computer algebra system in a real-world context.

Q: What are some key concepts related to finding $f(-3)$ in a real-world context?

A: Some key concepts related to finding $f(-3)$ in a real-world context include function evaluation, simplification of expressions, and rounding numbers to the nearest hundredth.

Q: Can I find $f(-3)$ using a specific mathematical technique or method in a real-world context?

A: Yes, you can find $f(-3)$ using a specific mathematical technique or method such as substitution, simplification, and evaluation in a real-world context.

Q: What are some common misconceptions about finding $f(-3)$ in a real-world context?

A: Some common misconceptions about finding $f(-3)$ in a real-world context include thinking that it's a simple arithmetic operation, not realizing the importance of following the order of operations, and not understanding the concept of rounding numbers