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Let A = [–1, 1]. discuss whether the following functions defined on A are one-one, onto or bijective: h(x) = x|x|

Let A = [–1, 1].  
 discuss whether the following functions defined on A are one-one, onto or bijective: 
 h(x) = x|x| 
 

Grade:12

1 Answers

Harshit Singh
askIITians Faculty 5964 Points
11 months ago
Dear Student


h (x) = x|x|
Let h (x1) = h (x2)
x1|x1| = x2|x2|
If x1,x2>0
x1^2= x2^2
x1^2–x2^2=0
(x1–x2)(x1+x2)=0
x1=x2(as x1+x2≠0)
Similarly for x1, x2< 0, we have x1= x2
It’s clearly seen thatfor x1and x2of opposite sign, x1≠x2.
Hence, f (x) is one-one.
For x ∈ [0,1]
f(x) = x^2∈[0,1]
For x<0,
f(x) = – x^2 ∈[-1,0)
Hence, the range is [-1, 1].
Thus, h (x) is onto.
Therefore, h (x) is bijective

Thanks

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