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Question

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

A
x+1,x-1
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B
-1+x,-1-x
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C
x+1,1-x
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D
x+1,-x-1
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Solution

The correct option is C x+1,1-x
Every linear function is either strictly increasing or strictly decreasing. If
f(x)=ax+b,
Df=[p,q]
Rf=[m,n]
then f(p)=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]
f(-1)=0 and f(1)=2
or f(-1) = 2 and f(1) = 0
Depending on f(x) is increasing or decreasing
a+b=0 and a+b=2 .......(i)
or a+b=2 and a+b=0 .......(ii)
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


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