# How do you find the domain and range of f(x)=10-x^2?

May 28, 2018

Domain = Real Number $\left(\mathbb{R}\right)$
Range = $\left(- \infty , 10\right]$

#### Explanation:

As $x$ can take any value so domain is real number.
For range We know that
${x}^{2} \ge 0$
So
$- {x}^{2} \le 0$
now add 10 on both side of equation
so equation become
$10 - {x}^{2} \le 10 + 0$
So the range is $\left(- \infty , 10\right]$

May 28, 2018

Domain: $x \in \mathbb{R}$
Range: f(x) in (-∞, 10]

#### Explanation:

Well, first, let's explain what a domain and range is.

A domain is the set of argument values (or "input") in which the function is defined. So, for example. for a function $g \left(x\right) = \sqrt{x}$, the domain will be all non-negative real numbers, or $x \ge 0$.

For this function $f \left(x\right)$, we see that the function has no square roots, fractions, or logarithmic functions that would be undefined for certain values of $x$.

Therefore, the domain of this function is all real numbers, or $x \in \mathbb{R}$.

The range of a function is all possible values (or "output") of the function, after substituting in the domain. So, for example, a function such as $h \left(x\right) = x$ will have the range as all real numbers, but a function such as $j \left(x\right) = \sin \left(x\right)$ can only output values in between -1 and 1, so the range is $\left[- 1 , 1\right]$, or $- 1 \le j \left(x\right) \le 1$.

To find the range of $f \left(x\right)$, we must first observe that the function has no minimum value. This can be done two ways.

First, we can observe that the coefficient in front of the ${x}^{2}$ term is negative. So, as $x$ increases (or decreases), ${x}^{2}$ increases, and the value of $f \left(x\right)$ decreases. Thus there must be a maximum value for $f \left(x\right)$, which is 10 in this case, when $x = 0$. You may have to complete the square, or use some other method for other functions.

Or, we can just see the graph of $y = f \left(x\right)$. graph{y = 10-x^2}
From the graph, it is clear that the maximum value of $f \left(x\right)$ is 10.

So, we can conclude that the domain of the function is all real numbers, or $\mathbb{R}$, and the range of the function is (-∞, 10] in set notation.

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