Let $f(x)$ be such that $f(x+2) = f(x)$ and $f(-x) = f(x)$ for any real number $x$. On the interval $[2,3]$, $f(x) = x$. Then the formula of $f(x)$ given on $[-2,0]$ is:
Step-by-Step Solution
Key Concept: Use the even function property f(-x) = f(x) and periodicity f(x+2) = f(x) to translate the known formula f(x) = x on [2,3] backwards to [-2,0]. On [2,3], apply even symmetry to get values on [-3,-2], then use periodicity to shift to [-1,0], and finally apply even symmetry again to determine the formula on [-2,-1].
Given $f(x) = x$ on $[2,3]$ with period 2, we use periodicity and even function properties: $f(x) = x+4$ on $[-2,-1]$ and $f(x) = x+2$ on $[0,1]$. Since $f(x) = 2-x$ on $[-1,0]$ is even, on $[-2,0]$ we have $f(x) = 3-|x+1|$.
Correct Answer: 3