Question

How do I find the range of the function `f(x)=10-x^2`?

Answer 1

The range of `f` is `(-infty,10]`.

Let us look at some details.

Since `x^2 ge 0`,

`f(x)=10-x^2 le 10-0=10`,

so the largest value of `f` is `10`.

Since `x^2` can be made as large as we wish by choosing large, `f(x)=10-x^2` can be made as small as we wish, which can be represented by `-infty`.

Hence, the range is `(-infty,10]`.

The graph of `f` looks like this:

I hope that this was helpful.