Question

A function $f$ has domain $[-1,2]$ and range $[0,1]$. The domain and range respectively of the function $g$ defined by $g\left(x\right)=1-f(x+1)$ is?

• A. $[-1,1];[-1,0]$
• B. $[-2,1];[0,1]$
• C. $[0,2];[-1,0]$
• D. $[1,3];[-1,0]$

Answer 1

The correct option is B $[-2,1];[0,1]$

Domain of $f\left(x\right)$ is $[-1,2]$ means $-1\le x\le 2$,

$g\left(x\right)=1-f(x+1)$

So domain of $g\left(x\right)$ will be defined by $f(x+1)$,

Now domain of $f(x+1)$ will be $-1\le x+1\le 2\to -2\le x\le 1$

Domain of $g\left(x\right):[-2,1]$

Range: $g\left(x\right)$ will be maximum when $f(x+1)$ is minimum as it is negative. Similarly $g\left(x\right)$ will be minimum when $f(x+1)$ is maximum.

$\therefore$ Min is $g\left(x\right)=1-\left(1\right)=0$

Max is $g\left(x\right)=1-\left(0\right)=1$

$\therefore$ Range: $[0,1]$

Hence, $\left(B\right)$