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.

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

 is a quadratic function, so its graph is a parabola. The key is to find the -coordinate of the vertex of the parabola, which can be found by completing the square:

The vertex occurs at , so the interval which contains this value will have at least one pair  such that . 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 .

 has a sinusoidal wave as its graph, with period ; it begins at a relative maximum of  and has a relative maximum or minimum every  units. Therefore, any interval containing an integer multiple of  will have at least two distinct values  such that .

The only interval among the choices that includes a multiple of  is :

 .

This is the correct choice.

We could solve this by actually adding up all terms the from 0 to 30, but it would take way too much time. There is a simple formula to remember for the summations of consecutive terms:  , which gives the sum of all terms from 0 to .

By substituting the value provided in our problem  into the formula, we can solve for the correct answer.