Question

What is the range of the function, `f(x) = 1- 1/(1+x^2)`?

Answer 1

Range is `(0,1]`.

Explanation:

As `x^2` can take value from `0` to `oo`,

`1/(1+x^2)` can take values from `1` to `0`

and `1-1/(1+x^2)` can take values from `01` to `1` i.e. range is `(0,1]`.

graph{1-1/(1+x^2) [-10, 10, -5, 5]}

Answer 2

The range is `f(x) in [0,1)`

Explanation:

Let `y=1-(1/(1+x^2))`

Determination of `x` in terms of `y`

`y=(1+x^2-1)/(1+x^2)=x^2/(1+x^2)`

`y(1+x^2)=x^2`

`x^2(1-y)=y`

`x^2=(y)/(1-y)`

`x=sqrt((y)/(1-y))`

In order for `x` to have a solution, solve this inequality

`y/(1-y)>=0`

Build a sign chart

`color(white)(aaaa)` `y` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `0` `color(white)(aaaaaaa)` `1` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `y` `color(white)(aaaaaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `1-y` `color(white)(aaaaaa)` `+` `color(white)(aaaaa)` `color(white)(a)` `+` `color(white)(aa)` `||` `color(white)(aa)` `-`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+` `color(white)(aaa)` `||` `color(white)(aa)` `-`

Therefore,

The range of y=f(x) is

`y in [0,1)`