Sets, Relations & Functions
Functions
star_batch_jee_advanced_2025
Grade 11
Question:
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:
x + 4
2 - x
3 - |x+1|
2 + |x+1|
Step-by-Step Solution
Key Concept: Periodicity and even/odd function properties can be combined to extend a function's definition across different intervals.
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