Rewrite this in  form.  The equation given is a parabola.

The negative ten coefficient tells us that the parabola will open downward.

The negative three indicates that the y-intercept is at .

Since the curve opens downward, there will be a maximum at the vertex, which is at , since the vertex  is zero as there is no coefficient for .

This means the range will start at negative infinity, and ends at negative three.

The answer is:  

The equation in slope-intercept form, , is:

This is a horizontal line, and the y-value will be fixed at this value, which means the range will not change.  Be careful not to misinterpret  or three as the slope!

The answer is:  

Recall that the domain of the function f(x) is the set of all x where f(x) is defined. Also, recall that two of the most common ways that a function can be undefined are division by zero and a negative inside of a square root. So, we need to ensure that are domain excludes all x values for which those things can occur. 

The  portion of our equation is of no concern, as the domain of  is all real numbers - that is to say, we can plug any real number into  and it will be defined. However, the  of our equation is cause for concern. We need to avoid two issues here - division by zero, and having a negative in the square root. We know that the only x value that would result in a zero in our denominator is zero, as the square root of zero is zero. That is, if we substitute zero for x, we will have  in our denominator, which evaluates to , leaving a zero in our denominator, leaving us with an undefined function. So, we need to exclude zero from our domain. Secondly, we need to exclude all number below zero from our domain as well, as x values below zero would result in a negative inside of a square root, which would result in an undefined function.

So, summarizing, we need to exclude all numbers below zero as well as zero itself from our domain, resulting in a final answer of: 

Recall that the domain of the function q(x) is the set of all x values that result in the function q(x) being defined. In this case, what we need to think about is avoiding a negative number inside of the radical (or square root symbol), which would result in an undefined function. What this means is that we need to all the x values for which the expression inside of the radical is greater than or equal to zero. In mathematical terms, we need to solve for x in the inequality:

Thus, we will have a positive or zero value inside of the radical only when x is less than or equal to -4 and greater than or equal to 4. If we look at the expression inside the radical, this makes sense - for all x values between -4 and 4,  will evaluate to a number less than 16, which will result in a negative inside of the radical and therefore an undefined function. For x values less than or equal to -4 and greater than or equal to 4,  will evaluate to a number greater than or equal to 16, which will result in a positive inside of the radical and therefore a defined function. Written in interval notation, this domain looks like:

The range of the parent function  is all real numbers.  The coefficient of the x cubed term will widen the curve, and flip the graph since it is a negative value.  The coefficient will not affect the range.

The negative ten at the end will only shift the graph downward 10 units and also will not affect the range of the function.

This means that the range includes all real numbers.

The answer is:  

The equation given is in the form of .

Since the value of , the location of the vertex  will be at .

Substitute  in order to determine the min or max of this curve.  Since the  coefficient is positive, the parabola will open upward, and will have a minimum.  The  coefficient will not affect the range.

The minimum point is .

The range is including all y-values that are existent on this graph.

The answer is:  

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 .