Question

There are exactly two distinct linear functions, which map [-1,1] on to [0,2], they are

• A. x+1,x-1
• B. -1+x,-1-x
• C. x+1,1-x
• D. x+1,-x-1

Answer 1

The correct option is C x+1,1-x

Every linear function is either strictly increasing or strictly decreasing. If
$f\left(x\right)=ax+b,$
$D_f=[p,q]$
$R_f=[m,n]$
then $f\left(p\right)=m$
f(q)=n
If f(x) is strictly increasing and f(q)=n,
f(q)=m if f(x) is strictly decreasing
Let f(x) = ax +b be the linear function which maps [-1, 1] on to [0, 2]
$\therefore$ f(-1)=0 and f(1)=2
or f(-1) = 2 and f(1) = 0
Depending on f(x) is increasing or decreasing
$\Rightarrow -a+b=0anda+b=2.......\left(i\right)$
or $-a+b=2anda+b=0‘.......\left(ii\right)$
Solving (i) we get a = 1, b = 1, solving (ii) we get a = -1, b = 1
Thus there are only two functions either x + 1 or -x + 1