If f(n) is a function such that f(1) = f(2) = f(3) = 1, and such that:
for n > 3, then f(6) is equal to:
- 2
- 3
- 7
- 11
- 26.
Big f(x), Find f(6)
for n > 3, then f(6) is equal to:
Identifying Patterns
for n > 3, then f(6) is equal to:
Find f(x+1) – f(x)
Hard Functional Equation, f(n) = n
The function f satisfies the functional equation
f(x) + f(y) = f(x + y) – xy – 1
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
Given Graphs,Find |f – g|
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)|.
Postage Function, Using [ ]
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.
Defined Operations 1; Do Composition
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?
Defined Operation; Do Inverse
Suppose a linear operation �/� is defined so that:
5 / = 11;
8 / = 17;
12 / = 25.
What is n if n / = 35?
Defined Operation; Find Formula
If 4 → 3, 10 → 6, 7 → 4.5, then what is the formula for x → ?
Progression to Quadratic
Consider the following interesting list of integers:
P0 = 41,
P1 = 43 = 41 + 2,
P2 = 47 = 43 + 4,
P3 = 53 = 47 + 6, … ,
P1 = 43 = 41 + 2,
P2 = 47 = 43 + 4,
P3 = 53 = 47 + 6, … ,
in which Pn is obtained by adding 2n to Pn-1.
It so happens that there is a quadratic function F(x) with the property that
Once you have the formula, consider the question of whether or not F(n) is a prime number for every nonnegative value of n.
Continued Fraction with 1’s and 3’s
What is the value of this fraction that continues forever in the same pattern? (You’ll want the quadratic formula.)