To solve the equation f(x) = 2f(x-4) + 18, we start by substituting the given value of f(1) which is 2.
First, let’s see what happens when we set x = 1 in the original equation:
- f(1) = 2f(1-4) + 18
- f(1) = 2f(-3) + 18
Since we know that f(1) = 2, we can substitute that value into the equation:
- 2 = 2f(-3) + 18
Now, let’s solve for f(-3):
- 2 – 18 = 2f(-3)
- -16 = 2f(-3)
- f(-3) = -8
Now we have found that f(-3) = -8. Next, we will find f(-3) using the original equation again by substituting x = -3:
- f(-3) = 2f(-3-4) + 18
- f(-3) = 2f(-7) + 18
- -8 = 2f(-7) + 18
Solve for f(-7):
- -8 – 18 = 2f(-7)
- -26 = 2f(-7)
- f(-7) = -13
So now we have f(-7) is -13. If we keep repeating this process, we can find a pattern, or we can treat it as a recursive function.
However, the question specifically asks for the value of f(1), which we’ve already established is 2. This shows that the function f evaluated at 1 is indeed:
- f(1) = 2