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

1) This one is the most straightforward of the solutions to explain. Simply plug in -1 for x. This means that:

f(-1) = 5*(-1)⁴ - 4(-1)³+ 3(-1)² - 2(-1) +1 = 5*1 - (4)(-1) + 3 - (2)(-1) +1 = 5+4+3+2+1 = 15.

2) What this sequence is saying is, to find the next member of the sequence, multiply the previous number by 2 and add 1. We can just do this for every number f(1) = 3, so f(2) = 2*f(1)+1 = 2*3+1 = 7, f(3) = 2*7 +1 = 15, f(4) = 2*15 +1 = 31, f(5) = 31*2+1 = 63 and so on. You can calculate the rest of the terms up to f(11), since it's just the same pattern.

A cheat to make sure you have the right answer: Because we are multiplying every number by 2 and adding 1 and the fact that we started with one less than a multiple of 2 means that f(n) = 2ⁿ⁺¹ - 1 (a formal proof would involve induction, but that's much higher level than an algebra class). In this case, f(11) = 2¹² - 1 = 4095.

3) This one is a bit of a doozy, so strap yourself in.

Since x = 1 ≤ 5, we go to the second condition.

This means that f(1) = 1 - f(f(1+3)) = 1 - f(f(4). (1)

Since 4 ≤ 5, we go to the second condition again to find out what f(4) is and thus what f(f(4)) is:

We get that f(4) = 4 - f(f(7)) (2)

Since 7 ≥ 5, we go to the first condition:

f(7) = 7 - 2 = 5.

Going back to (2), we get that f(4) = 4 - f(5). (3)

Since 5 ≥ 5, we go to the first condition:

f(5) = 5 - 2 = 3.

Plugging this into (3), we get that f(4) = 4 - 3 = 1

Finally, subbing this back into (1), we get that:

f(1) = 1 - f(1) which implies 0 = 1 - 2f(1), which means f(1) = 1/2.

4) This is similar to 2), but since we're only being asked to calculate out to f(4), it's compact enough for me to write out in full. f(1) = f(0)+2*1 = 0 + 2 = 2. f(2) = f(1) + 2*2 = 2 + 2*2 = 2+4 = 6, f(3) = f(2)+2*3 = 6+6 = 12, and finally f(4) = f(3)+2*4 = 12+8 = 20.

5) From our function we get that f(7) = f(7-2) + 7 = f(5)+7.

Since we already know that f(7) = 11, we just get the equation 11 = f(5) + 7 which means f(5) = 4.

6) Again from our function we know that f(1) + 2 f*(5-1) = 1, which means f(1) + 2*f(4) = 1 (4). We don't know what f(4) is, but we can try to find out.

From our function we know that: f(4) + 2*f(5-4) = 4.

This means that: f(4)+ 2*f(1) = 4 (5)

We have two linear equations: (4) and (5). Let's make f(1) = x, and f(4) = y. In this case, we are trying to solve for x.

We can rewrite our equations as follows: x + 2y = 1 and y + 2x = 4 (2x + y = 4)

We can solve this by eliminating a variable. y would be most convenient here, since we're interested in the value of x (f(1)):

x+2y = 1

-2(2x+y) = -2 (4), which means -4x - 2y = -8.

If we add these two new equations together to get an equivalent, more useful one, we get that

(x - 4x) + (2y - 2y) = -7, which implies -3x = -7, or x = f(1) = 7/3.

Answer 2

Duplicate question, disregard this question.