Question

What is the domain & range of the function : `x^2/(1+x^4)`?

what is the domain & range of the function : `f(x)=x^2/(1+x^4)`?

Answer 1

The domain is `(-oo, oo)` and the range `[0, 1/2]`

Explanation:

Given:

`f(x) = x^2/(1+x^4)`

Note that for any real value of `x`, the denominator `1+x^4` is non-zero.

Hence `f(x)` is well defined for any real value of `x` and its domain is `(-oo, oo)`.

To determine the range, let:

`y = f(x) = x^2/(1+x^4)`

Multiply both ends by `1+x^4` to get:

`y x^4 + y = x^2`

Subtracting `x^2` from both sides, we can rewrite this as:

`y (x^2)^2-(x^2)+y = 0`

This will only have real solutions if its discriminant is non-negative. Putting `a=y`, `b=-1` and `c=y`, the discriminant `Delta` is given by:

`Delta = b^2-4ac = (-1)^2-4(y)(y) = 1-4y^2`

So we require:

`1-4y^2 >= 0`

Hence:

`y^2 <= 1/4`

So `-1/2 <= y <= 1/2`

In addition note that `f(x) >= 0` for all real values of `x`.

Hence `0 <= y <= 1/2`

So the range of `f(x)` is `[0, 1/2]`