For n ∈ N, f n(x) denotes the composition of f with itself n times.Provide a formula (i.e. a ”rule”) for f n(x) when f (x) = −x + 2
Question
For n ∈ N, f n(x) denotes the composition of f with itself n times.
Provide a formula (i.e. a ”rule”) for f n(x) when
f (x) = −x + 2
Solution
The function f(x) = -x + 2 is a linear function that flips the sign of x and adds 2. When we compose this function with itself n times, we get f^n(x).
Let's start with n=1, we have f(x) = -x + 2.
For n=2, we have f(f(x)) = f(-x + 2) = -(-x + 2) + 2 = x + 2.
For n=3, we have f(f(f(x))) = f(x + 2) = - (x + 2) + 2 = -x.
For n=4, we have f(f(f(f(x)))) = f(-x) = -(-x) + 2 = x + 2.
From these calculations, we can see a pattern emerging. The function f^n(x) alternates between -x + 2 and x + 2 for odd and even n respectively.
So, we can write the rule for f^n(x) as:
f^n(x) = { -x + 2, if n is odd { x + 2, if n is even
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