This is the composition of two functions. By definition, .  To find the range of , we need to find the values of this function for each value in the domain of . Since , this is equivalent to evaluating  for each value in the range of , as follows:

Range value: 3 

Range value: 5

Range value: 8

Range value: 13

Range value: 21

The range of  on the set of range values of  - and consequently, the range of  - is the set . Of the choices given, only 1 is not in this set.

The domain of a function is the set of all values of the independent variable, , over which the function is defined. The first step is usually to find where the function is undefined. For a rational function this is always going to consist of points where the denominator is zero.

Find the roots of the denominator:

                                                   (1)

This is not one we can easily factor. Therefore, we should use the quadratic formula.

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Recall that for a quadratic of the form  the general form of the solution is, 

                                          (2) 

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The solution to equation (1) is, 

                                                    (3)

The two roots are: 

Now that we have the roots for the denominator we can construct the domain using interval notation. Use open parenthesis to exclude the two roots themselves from the domain. Also think of how the roots will split up the number line into three regions. 

The total domain of our function in interval notation is: 

Finding the Domain

The domain of a function is defined as the set of all valid input values of  overwhich the function is defined. The simple rule of thumb for rational functions is that all real numbers will work except for those in which denominator is zero since division by zero is not allowed.

Set the denominator to zero and solve for , 

The function is therefore defined everywhere except at . Therefore the domain expressed in interval notation is,

Note that the open parentheses indicate that  is not in the domain, but  may become arbitrarily close to  . 

Finding the Range 

The range of a function is defined as the set of all outputs spanning the domain. Finding the range can be achieved by finding the domain of the inverse function. First solve   for  to obtain the inverse function, 

Multiply both sides by , 

Distribute , 

Move all terms with  to one side of the equation, 

Factor and solve for 

The inverse function is therefore,

Find the domain of the inverse function, 

The range of  is the domain of , which is:

If you look at the plots for the function  (in blue) and  (in red and labeled as  in the figure) you can see the asymptotic behavior of as  approaches  and of  as  approaches .

THe notation  is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).

The original expression for f(x) is . We will take each x and substitute in the value of g(x), which is 2x-1.

We will now distribute the -2 to the 2x - 1.

We must FOIL the  term, because . 

Now we collect like terms. Combine the terms with just an x.

Combine constants.

The answer is .

Plug 16 in for .  

Add 9 to both sides.  

Take the square root of both sides. =

Final answer is  

This expression is the same as saying "take the answer of and plug it into ."

First, we need to find . We do this by plugging in for in .

Now we take this answer and plug it into .

We can find the value of by replacing with .

This is our final answer.

Total cost of the cab ride is going to equal the base fare ($4.50) plus an additional 10 cents per mile. This means the ride will always start off at $4.50. As the cab drives, the cost will increase by $0.10 each mile. This is represented as $0.10 times the number of miles. Therefore the total cost is:

The pre-question text provides us with all of the information required to complete this problem. 

We know that the total length of the walls is to be  ft.

We also know that we have a total of  walls and  walls.

With this, we can set up an equation and solve for .

Our equation will be with sum of all the walls set equal to the total length of the wall...

Remeber, we want  in terms of , which means our equation should look like

 something

  Subtract  on both sides

  Divide by  on both sides

   Simplify

     Answer!!!

The function is written in slope-intercept form, which means:

where:

= slope

= x value

= y-intercept

Therefore, the slope is

The flat rate of 29.99 does not change depending on months of service.  It is $29.99 no matter how long services are in use.  The monthy fee is directly related to the number of months the services are in use.