Question

Let $f:R\to R$ and $g:R\to R$ be mappings such that $f(g(x))$ is an injective mapping, then which of the following statement(s) is(are) correct?

• A. $g(x)$ must be an injective mapping.
• B. If $g(x)$ is surjective then $f(x)$ may not be injective mapping.
• C. If $g(x)$ is not surjective, then $f(x)$ may be injective or many-one mapping.
• D. $f(x)$ may not be an injective mapping.

Answer 1

Answer: (A), (C), (D)

(A) As, $f(g(x))$ is one-one so $g(x)$ must be one-one function, otherwise $f(g(x))$ will not be one-one function.
(B) If $f(x)$ is not one-one, then $f\left(a_1\right)=f\left(a_2\right)$, where $a_1\ne a_2$.

As, $g$ is onto, so $g\left(b_1\right)=a_1$ and $g\left(b_2\right)=a_2$.
As, $f(g(x))$ is one-one, so $f\left(g\left(b_1\right)\right)=f\left(a_1\right)$ and $f\left(g\left(b_2\right)\right)=f\left(a_2\right)$, which is not possible.
So, $f(x)$ must be one-one.
(C) Clearly, if $g(x)$ is not onto then $f(x)$ may be both one-one or many- one function.
(D) If $f(g(x))$ is one-one then $f(x)$ may be both one-one or many one function. Ans.]