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Question

Find the range of f(x)=x2(1+x2) (x real).

A
0y<1
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B
0y1
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C
y<1
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D
y<1
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Solution

The correct option is C 0y<1
Since for every real x, 1+x20, therefore x2(1+x2)is a real number for all real x.

Hence the domain of f is the set R of all real numbers. The range of f consists of all real numbers y such that f(x)=y for real x.

Now f(x)=y=x2(1+x2)=y=x2=y+yx2=x=y(1y)...

(1) Since x is real, we must have y(1y)0(y1)which is satisfied if 0y1

Hence range of f=y:y is real and 0y<1

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