We want to add the two functions and combine like terms. So we use the formula:\left( f+g\right)\left(x\right) = f\left(x\right) + g\left(x\right)

We also want to find the domain of f(x) and g(x) to help us determine the domain of \left(f+g\right)\left(x\right).

The domain of the resulting function will be the intersection of the initial two functions, in this case f and g since we are finding the sum or difference of f and g.

First we find the function \left(f+g\right)\left(x\right):

Next, we identify the domain of f\left(x\right) and g\left(x\right):

Domain of f\left(x\right): -\infty \lt x \lt \infty

Domain of g\left(x\right): -\infty \lt x \lt \infty

Finally, we find the domain of \left(f+g\right)\left(x\right) by taking the intersection of the domains of f\left(x\right) and g\left(x\right).

\text{Domain of }\left(f+g\right)\left(x\right) \text{: }-\infty \lt x \lt \infty

Notice that the degree of the resulting function is the maximum of the degrees of the two functions.

We want to subtract the one function from the other function and combine like terms. So we use the formula:\left( f-g\right)\left(x\right) = f\left(x\right) - g\left(x\right)

From part (a) we know the domain of f\left(x\right), g\left(x\right), and the intersection of their domains (which is the domain of \left(f-g\right)\left(x\right)).

\text{Domain of } \left(f-g\right)\left(x\right)\text{: }-\infty \lt x \lt \infty

We want to multiply the two functions and combine like terms if possible. So initially we start with:

\left( f \cdot g\right) \left(x\right)= f\left(x\right)\cdot g\left(x\right)

We also want to find the domain of the resulting function and check that domain of the expression formed by \left(f \cdot g\right)\left(x\right).

Find the domain by taking the intersection of f(x), g(x), and the expression 7x^3-2x^2+21x-6.

Domain of the expression 7x^3-2x^2+21x-6: -\infty \lt x \lt \infty

\text{Domain of } \left(f \cdot g\right)\left(x\right)\text{: }-\infty \lt x \lt \infty

Notice that the degree of the resulting function is the sum of the degrees of f and g.

We want to divide one function by another function and combine like terms if possible. So initially we start with:

\left( \dfrac{f}{g}\right) \left(x\right)= \dfrac{f\left(x\right)}{ g\left(x\right)}

We also want to find the domain of the resulting function. We do this by checking the domain of the expression formed by \left(\dfrac{f }{ g}\right)\left(x\right), and excluding points where g(x)=0 (as we cannot divide by 0).

We need to work out where g(x)=0 so we can exclude those values from the domain.

g(x) = 7x -2 = 0 \implies x = \dfrac{2}{7}

\text{Domain of }\left( \dfrac{f}{g}\right) \left(x\right)\text{: } -\infty \lt x < \dfrac{2}{7} \cup \dfrac{2}{7} < x \lt \infty

The total cost of everyone would be \text{Total cost of everyone}=\text{Cost for hotel room}+\left(\text{theme park cost per person}\right)\cdot\left( \text{number of people}\right)The cost per person will be \text{Cost per person}=\dfrac{\text{Total cost of everyone}}{\text{Number of people}}

A function which represents the total cost of everyone would be: T(x)=280+60x

A quotient to determine the cost per person would be: C(x)=\dfrac{280+60x}{x}

We could rewrite this as:

We need to consider what x is representing and any constraints based on the context or restrictions algebraically.

Algebraically, since we have x in the denominator, we know that x\neq 0.

Based on the context, x is representing the number of people, so must be positive integer values. Also, there will be restrictions on the number of people that can stay in one hotel room. This will likely vary based on the hotel, but is generally around 4.

A possible domain could be x \in \left\{1, 2, 3, 4 \right\} if we assume a limit of 4 people per room.