Question

If $f\left(x\right)=\left(h_1\right(x)-h_1(-x\left)\right)\left(h_2\right(x)-h_2(-x\left)\right)\cdots \left(h_{2n+1}\right(x)-h_{2n+1}(-x\left)\right),$ where $h_1\left(x\right),h_2\left(x\right),\cdots \cdots h_n\left(x\right)$ are defined everywhere and $f\left(200\right)=0$, then $f\left(x\right)$ is

• A. one - one
• B. many one
• C. odd
• D. even

Answer 1

Answer: (B), (C)

The correct options are
B

many one

C

odd

$f(-x)=\left(h_1\right(-x)-h_1(x\left)\right)\left(h_2\right(-x)-h_2(x\left)\right)......\left(h_{2n+1}\right(-x)-h_{2n+1}(x\left)\right)$

$f(-x)=(-1{)}^{2n+1}f(x)$

$f\left(x\right)+f(-x)=0\Rightarrow f\left(x\right)$ is odd

$f(-200)=-f\left(200\right)=0$

$\Rightarrow f\left(x\right)$ is many one