If F ( X ) = 3 − 2 X F(x)=3-2x F ( X ) = 3 − 2 X And G ( X ) = 1 X + 5 G(x)=\frac{1}{x+5} G ( X ) = X + 5 1 ​ , What Is The Value Of ( F G ) ( 8 \left(\frac{f}{g}\right)(8 ( G F ​ ) ( 8 ]?A. -169 B. -1 C. 13 D. 104

If $f(x)=3-2x$ and $g(x)=\frac{1}{x+5}$, what is the value of $\left(\frac{f}{g}\right)(8)$?

Understanding the Problem

To find the value of $\left(\frac{f}{g}\right)(8)$, we need to first understand what the notation $\left(\frac{f}{g}\right)(x)$ means. This notation represents the quotient of two functions, $f(x)$ and $g(x)$. In other words, it represents the function that results from dividing $f(x)$ by $g(x)$.

Finding the Quotient of Two Functions

To find the quotient of two functions, we need to divide the function $f(x)$ by the function $g(x)$. This can be done by dividing the numerator of $f(x)$ by the denominator of $g(x)$.

Calculating the Quotient

Let's calculate the quotient of $f(x)$ and $g(x)$.

$$\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{3-2x}{\frac{1}{x+5}}$$

To simplify this expression, we can multiply the numerator by the reciprocal of the denominator.

$$\left(\frac{f}{g}\right)(x) = (3-2x)(x+5)$$

Now, let's expand the expression.

$$\left(\frac{f}{g}\right)(x) = 3x + 15 - 2x^2 - 10x$$

Combine like terms.

$$\left(\frac{f}{g}\right)(x) = -2x^2 - 7x + 15$$

Evaluating the Quotient at $x=8$

Now that we have the quotient of $f(x)$ and $g(x)$, we can evaluate it at $x=8$.

$$\left(\frac{f}{g}\right)(8) = -2(8)^2 - 7(8) + 15$$

First, let's calculate the square of 8.

$$(8)^2 = 64$$

Now, let's substitute this value back into the expression.

$$\left(\frac{f}{g}\right)(8) = -2(64) - 7(8) + 15$$

Next, let's calculate the product of -2 and 64.

$$-2(64) = -128$$

Now, let's substitute this value back into the expression.

$$\left(\frac{f}{g}\right)(8) = -128 - 7(8) + 15$$

Next, let's calculate the product of 7 and 8.

$$7(8) = 56$$

Now, let's substitute this value back into the expression.

$$\left(\frac{f}{g}\right)(8) = -128 - 56 + 15$$

Finally, let's calculate the sum of -128 and -56.

$$-128 - 56 = -184$$

Now, let's substitute this value back into the expression.

$$\left(\frac{f}{g}\right)(8) = -184 + 15$$

Finally, let's calculate the sum of -184 and 15.

$$-184 + 15 = -169$$

Conclusion

Therefore, the value of $\left(\frac{f}{g}\right)(8)$ is -169.

Answer

The correct answer is A. -169.
Q&A: If $f(x)=3-2x$ and $g(x)=\frac{1}{x+5}$, what is the value of $\left(\frac{f}{g}\right)(8)$?

Frequently Asked Questions

Q: What is the quotient of two functions?

A: The quotient of two functions is a new function that results from dividing one function by another. It is denoted by $\left(\frac{f}{g}\right)(x)$.

Q: How do I find the quotient of two functions?

A: To find the quotient of two functions, you need to divide the numerator of the first function by the denominator of the second function.

Q: What is the formula for the quotient of two functions?

A: The formula for the quotient of two functions is $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$.

Q: How do I simplify the quotient of two functions?

A: To simplify the quotient of two functions, you can multiply the numerator by the reciprocal of the denominator.

Q: What is the value of $\left(\frac{f}{g}\right)(8)$?

A: To find the value of $\left(\frac{f}{g}\right)(8)$, you need to substitute $x=8$ into the formula for the quotient of the two functions.

Q: How do I evaluate the quotient of two functions at a specific value of x?

A: To evaluate the quotient of two functions at a specific value of x, you need to substitute the value of x into the formula for the quotient of the two functions and simplify the expression.

Q: What is the final answer to the problem?

A: The final answer to the problem is -169.

Additional Resources

• For more information on the quotient of two functions, please see the article on "Quotient of Two Functions".
• For more information on how to simplify expressions, please see the article on "Simplifying Expressions".

Conclusion

In this article, we have discussed the quotient of two functions and how to find the value of $\left(\frac{f}{g}\right)(8)$. We have also provided a step-by-step solution to the problem and answered some frequently asked questions. We hope that this article has been helpful in understanding the concept of the quotient of two functions and how to apply it to solve problems.

Final Answer

The final answer to the problem is -169.