If *f*(*x*) = 4^{x}, then *f*(*x* + 1) – *f*(*x*) =

- 4
*f*(*x*)- 2
*f*(*x*) - 3
*f*(*x*) - 4
*f*(*x*)

The function *f* satisfies the functional equation

for every pair *x*, *y* of real numbers. If *f*(1) = 1, then the number of positive integers *n* for which *f*(*n*) = *n* is:

- 0
- 1
- 2
- 3
- infinite

Consider the numbers F_{n} defined by the following formula:

where *n* may be any positive integer. Calculate F_{1} through F_{5}. Do you notice anything suprising?

Suppose that the functions *f(x)* and *g(x)* have the following graphs:

Find the graph which most resembles the graph of |*f(x) – g(x)*|.

If the postal rate for first class letters is 42 cents for the first ounce or any portion thereof, and 17 cents for every ounce or portion thereof after the first ounce, then find the cost of a letter weighing *w* ounces, where *w* is some positive real number. Use [*x*] to stand for the greatest integer in *w*, as usual.

If 4 → 3, 10 → 6, 7 → 4.5, then what is the formula for *x* → ?

Suppose a linear operation �/� is defined so that:

5 / = 11;

8 / = 17;

12 / = 25.

What is *n* if *n* / = 35?

Suppose the operations “#” and “*t*” are defined so that:

6 # = 20; 2 # = 4; 10 # = 36; and 5 *t* = 4.5; 10 *t* = 7; 8 *t* = 6.

Then what is *n* if 3 # # *t* = *n*?

Consider the following interesting list of integers:

in which *P*_{n} is obtained by adding 2*n* to *P*_{n-1}.

It so happens that there is a quadratic function *F*(*x*) with the property that *P*_{n} = *F*(*n*)*n*. Find a formula for *F*(*x*).

Once you have the formula, consider the question of whether or not *F*(*n*) is a prime number for every nonnegative value of *n*.

What is the value of this fraction that continues forever in the same pattern? (You’ll want the quadratic formula.)