Question

Let $f : [0, 1] \to R$ be an injective continuous function that satisifes the condition $-1 < \, f (0) <\,f (1) <\,1$ Then, the number of functions $g : [-1, 1] \to [0, 1]$ such that $(gof) (x) =x$ for all $x \in\left[0,1\right]$ is

• A. 0
• B. 1
• C. more than 1, but finite
• D. infinite

Answer 1

Answer: (D)

We have,
$-1<\,f (0)<\,f (1)<\,1$
$g=[-1, 1] \to [0,1]$
go $f (x)=x, \forall \,x \, \in [0, 1]$
Only condition that $g (x)$ should satisfy for go $f (x)=x, \forall \,x \, \in [0, 1]$ is that g(x) should attain all values in $[0,1]$ when range of $f (x)$ a subset of $(-1,1)$ is used as image for $g(x)$. Thus, there can be infinite such functions $g(x)$ with domain $[-1,1]$ and range $[0, 1]$