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

Jul 22, 2016

$y \in \left(- \infty , 10\right]$

#### 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 $\mathbb{R}$.

Now, the square root of a number is always a positive number when working in $\mathbb{R}$. 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.

$\textcolor{p u r p \le}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{x}^{2} \ge 0 \textcolor{w h i t e}{a} \left(\forall\right) x \in \mathbb{R}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

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 $x \in \mathbb{R} \text{\} \left\{0\right\}$ 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]}