Let $f(x) = 8x + 6$ And $g(x) = 8 - X$. Find The Following:(a) $(f+g)(x$\](b) $(f-g)(x$\](c) $(f \cdot G)(x$\](d) $\frac{f}{g}(x$\](e) The Domain Of $\frac{f}{g}$---Answers:(a) $(f+g)(x) =

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(a) $(f+g)(x)$

To find the sum of two functions, we need to add their corresponding terms. In this case, we have:

$(f+g)(x) = f(x) + g(x)$

$= (8x + 6) + (8 - x)$

$= 8x + 6 + 8 - x$

$= 8x - x + 6 + 8$

$= 7x + 14$

Therefore, the sum of the two functions is $(f+g)(x) = 7x + 14$.

(b) $(f-g)(x)$

To find the difference of two functions, we need to subtract their corresponding terms. In this case, we have:

$(f-g)(x) = f(x) - g(x)$

$= (8x + 6) - (8 - x)$

$= 8x + 6 - 8 + x$

$= 8x + x + 6 - 8$

$= 9x - 2$

Therefore, the difference of the two functions is $(f-g)(x) = 9x - 2$.

(c) $(f \cdot g)(x)$

To find the product of two functions, we need to multiply their corresponding terms. In this case, we have:

$(f \cdot g)(x) = f(x) \cdot g(x)$

$= (8x + 6) \cdot (8 - x)$

$= 8x \cdot (8 - x) + 6 \cdot (8 - x)$

$= 64x - 8x^2 + 48 - 6x$

$= -8x^2 + 64x - 6x + 48$

$= -8x^2 + 58x + 48$

Therefore, the product of the two functions is $(f \cdot g)(x) = -8x^2 + 58x + 48$.

(d) $\frac{f}{g}(x)$

To find the quotient of two functions, we need to divide their corresponding terms. In this case, we have:

$\frac{f}{g}(x) = \frac{f(x)}{g(x)}$

$= \frac{8x + 6}{8 - x}$

Therefore, the quotient of the two functions is $\frac{f}{g}(x) = \frac{8x + 6}{8 - x}$.

(e) The domain of $\frac{f}{g}$

The domain of a function is the set of all possible input values for which the function is defined. In this case, we have:

$\frac{f}{g}(x) = \frac{8x + 6}{8 - x}$

For this function to be defined, the denominator $8 - x$ must be non-zero. Therefore, we must have:

$8 - x \neq 0$

Solving for $x$, we get:

$x \neq 8$

Therefore, the domain of the function is all real numbers except $x = 8$.

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(a) $(f+g)(x)$

Q: What is the sum of the two functions $f(x)$ and $g(x)$?

A: The sum of the two functions is $(f+g)(x) = 7x + 14$.

(b) $(f-g)(x)$

Q: What is the difference of the two functions $f(x)$ and $g(x)$?

A: The difference of the two functions is $(f-g)(x) = 9x - 2$.

(c) $(f \cdot g)(x)$

Q: What is the product of the two functions $f(x)$ and $g(x)$?

A: The product of the two functions is $(f \cdot g)(x) = -8x^2 + 58x + 48$.

(d) $\frac{f}{g}(x)$

Q: What is the quotient of the two functions $f(x)$ and $g(x)$?

A: The quotient of the two functions is $\frac{f}{g}(x) = \frac{8x + 6}{8 - x}$.

(e) The domain of $\frac{f}{g}$

Q: What is the domain of the function $\frac{f}{g}$?

A: The domain of the function is all real numbers except $x = 8$.

Additional Questions and Answers

Q: What is the value of $(f+g)(x)$ when $x = 2$?

A: To find the value of $(f+g)(x)$ when $x = 2$, we need to substitute $x = 2$ into the equation $(f+g)(x) = 7x + 14$. This gives us:

$(f+g)(2) = 7(2) + 14$

$= 14 + 14$

$= 28$

Therefore, the value of $(f+g)(x)$ when $x = 2$ is $28$.

Q: What is the value of $(f-g)(x)$ when $x = 3$?

A: To find the value of $(f-g)(x)$ when $x = 3$, we need to substitute $x = 3$ into the equation $(f-g)(x) = 9x - 2$. This gives us:

$(f-g)(3) = 9(3) - 2$

$= 27 - 2$

$= 25$

Therefore, the value of $(f-g)(x)$ when $x = 3$ is $25$.

Q: What is the value of $(f \cdot g)(x)$ when $x = 4$?

A: To find the value of $(f \cdot g)(x)$ when $x = 4$, we need to substitute $x = 4$ into the equation $(f \cdot g)(x) = -8x^2 + 58x + 48$. This gives us:

$(f \cdot g)(4) = -8(4)^2 + 58(4) + 48$

$= -8(16) + 232 + 48$

$= -128 + 232 + 48$

$= 152$

Therefore, the value of $(f \cdot g)(x)$ when $x = 4$ is $152$.

Q: What is the value of $\frac{f}{g}(x)$ when $x = 5$?

A: To find the value of $\frac{f}{g}(x)$ when $x = 5$, we need to substitute $x = 5$ into the equation $\frac{f}{g}(x) = \frac{8x + 6}{8 - x}$. This gives us:

$\frac{f}{g}(5) = \frac{8(5) + 6}{8 - 5}$

$= \frac{40 + 6}{3}$

$= \frac{46}{3}$

Therefore, the value of $\frac{f}{g}(x)$ when $x = 5$ is $\frac{46}{3}$.

Conclusion

In this article, we have found the sum, difference, product, and quotient of two functions $f(x) = 8x + 6$ and $g(x) = 8 - x$. We have also found the domain of the function $\frac{f}{g}$. Additionally, we have answered several questions and provided examples to illustrate the concepts.