Question

What is the range of the function `f(x)=10-x^2`?

Answer 1

`y in (-oo, 10]`

Explanation:

The range of a function represents all the possible output values that you can get by plugging in all the possible `x` values allowed by the function's domain.

In this case, you have no restriction on the domain of the function, meaning that `x` can take any value in `RR`.

Now, the square root of a number is always a positive number when working in `RR`. This means that regardless of the value of `x`, which can take any negative values or any positive value, including `0`, the term `x^2` will always be positive.

`color(purple)(|bar(ul(color(white)(a/a)color(black)(x^2 >=0 color(white)(a)(AA) x in RR)color(white)(a/a)|)))`

This means that the term

`10 - x^2`

will always be smaller than or equal to `10`. It will be smaller than `10` for any `x in RR "\"{0}` and equal to `10` for `x=0`.

The range of the function will thus be

`color(green)(|bar(ul(color(white)(a/a)color(black)(y in (- oo, 10]color(white)(a/a)|)))`

graph{10 - x^2 [-10, 10, -15, 15]}