Given the functions:

$f(x) = 7x + 4$ $g(x) = 1 - 6x$

We need to find the following expressions:

1. $(f + g)(x)$
2. $(f - g)(x)$
3. $(f \cdot g)(x)$
4. $(f \cdot f)(x)$
5. $\left( \frac{f}{g} \right)(x)$
6. $\left( \frac{g}{f} \right)(x)$

1. $(f + g)(x)$

To add the two functions, simply add $f(x)$ and $g(x)$ together:

$(f + g)(x) = f(x) + g(x) = (7x + 4) + (1 - 6x)$ Combine like terms:

$(f + g)(x) = 7x - 6x + 4 + 1 = x + 5$

2. $(f - g)(x)$

To subtract $g(x)$ from $f(x)$, subtract the expression for $g(x)$ from $f(x)$:

$(f - g)(x) = f(x) - g(x) = (7x + 4) - (1 - 6x)$ Distribute the negative sign and combine like terms:

$(f - g)(x) = 7x + 4 - 1 + 6x = 13x + 3$

3. $(f \cdot g)(x)$

To multiply the two functions, simply multiply $f(x)$ by $g(x)$:

$(f \cdot g)(x) = f(x) \cdot g(x) = (7x + 4)(1 - 6x)$ Use the distributive property (FOIL):

$(f \cdot g)(x) = 7x(1) + 7x(-6x) + 4(1) + 4(-6x) = 7x - 42x^2 + 4 - 24x$ Combine like terms:

$(f \cdot g)(x) = -42x^2 - 17x + 4$

4. $(f \cdot f)(x)$

To multiply $f(x)$ by itself, square $f(x)$:

$(f \cdot f)(x) = f(x) \cdot f(x) = (7x + 4)(7x + 4)$ Use the distributive property (FOIL):

$(f \cdot f)(x) = 7x(7x) + 7x(4) + 4(7x) + 4(4) = 49x^2 + 28x + 28x + 16$ Combine like terms:

$(f \cdot f)(x) = 49x^2 + 56x + 16$

5. $\left( \frac{f}{g} \right)(x)$

To divide $f(x)$ by $g(x)$, we take the ratio:

$\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{7x + 4}{1 - 6x}$

This is the simplified form of the expression.

6. $\left( \frac{g}{f} \right)(x)$

To divide $g(x)$ by $f(x)$, we take the ratio:

$\left( \frac{g}{f} \right)(x) = \frac{g(x)}{f(x)} = \frac{1 - 6x}{7x + 4}$

This is the simplified form of the expression.

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Summary of Results:

1. $(f + g)(x) = x + 5$
2. $(f - g)(x) = 13x + 3$
3. $(f \cdot g)(x) = -42x^2 - 17x + 4$
4. $(f \cdot f)(x) = 49x^2 + 56x + 16$
5. $\left( \frac{f}{g} \right)(x) = \frac{7x + 4}{1 - 6x}$
6. $\left( \frac{g}{f} \right)(x) = \frac{1 - 6x}{7x + 4}$

Would you like any further details or explanations on these calculations?

Here are some related questions for further practice:

1. How would the graph of $(f + g)(x)$ compare to the individual graphs of $f(x)$ and $g(x)$?
2. What happens to $\left( \frac{f}{g} \right)(x)$ as $x$ approaches certain values?
3. How can you find the roots of the equation $(f \cdot g)(x) = 0$?
4. What are the domain restrictions for $\left( \frac{f}{g} \right)(x)$?
5. How would you solve $(f \cdot f)(x) = 0$?

Tip: When adding or subtracting functions, always combine like terms (e.g., constants with constants, and terms with the same variable power).