by definition.

  is piecewise defined, with one defintion for negative values of the domain and one for nonnegative values. 

If , then we use the definition . This happens if

 or 

Therefore, the defintion of   for  or  is

Subsquently, if , we use the defintion , since :

.

The correct choice is

This can be understood better by substituting , and, subsequently,  in the function's definition.

which is now in standard quadratic form in terms of .

Write this in vertex form by completing the square:

Substitute  back for , and the original function can be rewritten as

.

To find the range, note that . Therefore, 

and 

The range of  is the set .

This can be understood better by substituting , and, subsequently,  in the function's definition.

which is now in standard quadratic form in terms of .

Write this in vertex form by completing the square:

Substitute  back for . The original function can be rewritten as

or, in radical form,

 can assume any real value; so, subsequently, can . But its square must be nonnegative, so

and 

The range of  is 

In order to find exactly the  values where the equations intersect and when . We need to consider each piece of information seperately.

Let's start with . Plugging  into , we have . Plugging 0 into this, we have. This in turn equals 1, because we were given that piece of information in the beginning. So we end up with

Now let's shift our attention to "intersect only when " That means, if we plug 1 into both equations, we can set them equal to each other.

becomes becomes .

Now we have two different equations arising from the two previous paragraphs.

We can solve this system of equations using the substitution method.

Solving for  in the first equation gives .

Plugging this equation in for  the 2nd equation gives . Using algebra on this equation we get

Now we are ready to use the quadratic formula to solve for .

Finally, since we're told in the beginning that , we must pick the plus sign in our solution for . Hence

.

This question is asking us to find the composition of f and g. In order to do this we need to plug g(x) into the x value in f(x).

A Fibonacci-style sequence starts with two numbers, with each successive number being the sum of the previous two. The second term is therefore the difference of the fourth and third terms, and the first term is the difference of the third and second.

Second term: 

First term: 

This is not among the given choices.

 has an inverse on a given domain if and only if there are no two distinct values on the domain  such that .

The key to this question is to find the zeroes of the polynomial, which can be done as follows:

The zeroes are , or approximately .

The polynomial being cubic, its graph has two vertices; since all three zeroes are in the interval , so are both vertices. Therefore, this interval must have at least one pair  such that . Since a cubic polynomial has two "arms", one going up and one going down,  will continuously increase or decrease over the other intevals. The correct choice is .

 has an inverse on a given domain if and only if there are no two distinct values on the domain  such that .

If , then  can be defined to be .

This happens if 

, or

Similarly, if ,  can be defined to be , or .

Either way, on any interval that does not include the value , the function can be restated as a linear function, which must have an inverse on that domain. 

On the one interval that does contain this value, , two values  can be found such that . For example, 

 is the correct response.

 has an inverse on a given domain if and only if there are no two distinct values on the domain  such that .

 is a constantly increasing function, since it the equation of a line with positive slope. The cube root of a constantly increasing function is also constantly increasing, so 

 always increases as  increases.

Therefore, if , .  has an inverse on  and, subsequently, any domain.