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Find the range of the following functions. (a) `f(x)=5-7x " (b) "f(x)=5-x^(2)` (c ) `f(x)=(x^(2))/(x^(2)+1)` |
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Answer» Correct Answer - (a) `R " (b) " (-oo,5] " (c ) "[0,1)` (a) `f(x)=5-7x" or " y=5-7x. "So, "x=(5-y)/(7).` Clearly, for any real value fo y there exists a real number x. So, range is set R. (b) `f(x)=5-x^(2)" or "y=5-x^(2). " So "x^(2)=5-y.` Since, `x^(2) ge 0, 5-y ge 0. ` So, `y le 5`. Therefore, range is `(-oo,5]`. (c ) `f(x)=(x^(2))/(x^(2)+1)=y.` ` :. x^(2)=yx^(2)+yimpliesx^(2)=(y)/(1-y).` Since `x^(2) ge 0` for all real `x,(y)/(1-y) ge " or " (y)/(y-1) le 0.` So, `0le y lt 1.` Therefore, range is `[0,1)` |
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