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)`

Answer

(a) `R " (b) " (-oo,5] " (c ) "[0,1)`

Explanation

(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)`