what's the domain and range of f(x)=4/x squared+1 also the x and y intercepts and asymptotes ? As well how do I graph this function?

\frac{4}{x^{2}+1}

The domain is what you put in and the range is what comes out.

For the domain, ask yourself, what values can I enter for x and get a valid result?
Do any entries result in a non-real solutions? Do any entries result in a negative in a square root? Those are normally not viable solutions.

As for the asymptotes

horizontal: if the power of the numerator is less than the power of the denominator, then the x-axis is a horizontal asymptote.

vertical: Are there any values which make the denominator 0? If not, then there is no vertical asymptote.

For the y-intercept, set x=0 and see what you get. That's the y-intercept.

For the x-intercepts, the horizontal asymptote will answer that.

Graph and you can see it.