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NCERT Class XI Mathematics - Relations and Functions - Solutions

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Question : 30 of 36
Marks: +1, -0
Let f = {(x,x21+x2):x∈R}\{ \left(x, \frac{x^2}{1+x^2}\right) : x \in R \} be a function from R into R. Determine the range of f.
Solution:  
We have f = {(x,x21+x2):x∈R}\{ \left(x, \frac{x^2}{1+x^2}\right) : x \in R \}
Clearly, domain of f is R.
Let y = x21+x2\frac{x^2}{1+x^2}
It is clear that x21+x2\frac{x^2}{1+x^2} ≥ 0 (Since x2x^2 ≥ 0 and 1 + x2x^2 ≥ 0)
and x2x^2 < 1 + x2x^2 ⇒ x21+x2\frac{x^2}{1+x^2} < 1
⇒ 0 ≤ y < 1
Hence, range of f is [0, 1).
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