Question

Which of following statements is/are INCORRECT?
I. If $f\left(x\right)$ and $g\left(x\right)$ are one-one, then $f\left(x\right)+g\left(x\right)$ is also one-one.
II. If $f\left(x\right)$ and $g\left(x\right)$ are one-one, then $f\left(x\right).g\left(x\right)$ is also one-one.
III. If $f\left(x\right)$ is odd, then it is necessarily one to one.

• A. I and II only
• B. II and III only
• C. III and I only
• D. I , II and III

Answer 1

Answer: (D)

I , II and III

Explanation through examples:
I. Let's assume $f\left(x\right)=x$ and $g\left(x\right)=-x$.
Both $f\left(x\right)$ and $g\left(x\right)$ are one-one, but $f\left(x\right)+g\left(x\right)=0$ is not one-one.
$\therefore$ Statement I does not hold true.
II. Let's assume $f\left(x\right)=x$ and $g\left(x\right)=x^3$.
Both $f\left(x\right)$ and $g\left(x\right)$ are one-one, but $f\left(x\right)\cdot g\left(x\right)=x^4$ is not one-one.
$\therefore$ Statement II does not hold true.
III. Let's assume $f\left(x\right)=\sin x$.
Here, $f\left(x\right)$ is an odd function, but it is not one-one.
$\therefore$ Statement III does not hold true.
$\therefore$ All three statements are incorrect.