Question

Let $f : [0, 1] \to [-1, 1]$ and $g : [-1, 1] \to [0, 2]$ be two functions such that $g$ is injective and $g$ of ; $[0,1]\to [0,2]$ is surjective Then,

• A. f must be injective but need not be surjective
• B. f must be surjective but need not be injective
• C. f must be bijective
• D. f must be a constant function

Answer 1

Answer: (B)

Let $h (x)=g (f (x))$ where, $g(x)$ is injective and $h(x)$ is surjective $h(x)$ is surjective
$\therefore $ Codomain of $h(x)$ = Range of $h(x)$.
Range of $h (x) = [0, 2]$ which is also codomain of $g$.
So, must be surjective.
Now, domain of $g=[-1, 1]$ which must be range of $f$.
But codomain of $f=[-1,1]$
So, it must be surjective