Solve for the following problems. Provide a detailed solution.

Question

1. If f(x) = 5x^4 - 4x^3 + 3x^2 - 2x + 1, find f(-1).

2. If f(n+1) = 2f(n) + 1 for every natural number n and f(1) = 3, find f(11).

3. The function is called a nested function, and is defined as:

f(x) = {x - 2 for x > 5

{x - f(f(x + 3)) for x <= 5

Find f(1).

4. A function is defined by f(0) = 0 and f(n) = f(n - 1) + 2n, for n>0. Find f(4).

5. If (x) = f(x - 2) + x, and f(7) = 11, find f(5).

6. Function f satisfies f(x) + 2f(5 - x) = x for all real numbers x. What is the value of f(1) ?

Answer 1

Hi Victor B.,

It's not the purpose of Wyzant to do your homework sets for you; in fact, that's quite discouraged!

Looks like q3. might pose the greatest structural challenge to solution, so I'll run through it in detail. You want to solve for f(1).

Since x is <=5, the 2nd portion of definition applies:

f(1) = {1-f(f(4))} following the nested definition down another level:

={1-f({4-f(f(7))}} now we may apply the 1st part of the definition:

={1-f({4-f(5)}} this pops out with the 2nd portion of definition again:

={1-f({4-{5-f(f(8))}})} apply part 1:

={1-f({4-{5-f(7)}})} apply part 1:

={1-f({4-{5-5}})} reduce:

={1-f(4)} apply part 2:

={1-{4-f(f(7))}} apply part 1:

={1-{4-f(5)}} apply part 2:

={1-{4-{5-f(f(8))}}} apply part 1:

={1-{4-{5-f(7)}}} apply part 1:

={1-{4-{5-5}}} reduce:

=-3

So, work the rest of your question parts out in similar fashion!

--Cheers, --Mr. d.