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Let f ={(x ,(x^2)/(1+x^2)):x in R}be a f...

Let `f ={(x ,(x^2)/(1+x^2)):x in R}`be a function from R into R. Determine the rage of f.

Text Solution

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The correct Answer is:
N/a
`f(x)=(x^(2))/(1+x^(2))`
Domain-f(x), is defined for all real values of x.
`therefore` Domain of f(x) = R
Range, let `y=(x^(2))/(1+x^(2))`
`impliesy+yx^(2)=x^(2)impliesy=x^(2)(1-y)`
`x^(2)=(y)/(1-y)`
`impliesx=+-sqrt((y)/(1-y))=+- sqrt(y(1-y))/(1-y),yne1`
which is difined if y `(1-y)ge0`
`{:(implies,yge0","1-yge0oryle0","1-yge0),(implies,yge0","yle1 oryle0","yge1),(Now",",yle0","yle1implies1 "is impossible"","and y ne1):}`
`thereforeyge0,yle1impliesy in[0,1[`
Therefore, range of f(x) = [0,1[.
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