# How do you find the domain and range of y=-(x-2)^2 +3 ?

Apr 27, 2017

Domain: $\left(- \setminus \infty , \setminus \infty\right)$
Range: $\left(- \setminus \infty , 3\right]$

#### Explanation:

To find the domain we have to check for divide by zeroes and negatives under radicals.

Since there are no fractions, divide by zero is impossible. Since we aren't any radicals ($\setminus \sqrt{x}$), it is impossible for that to happen. So we can conclude that we can put any value of x into the function and get an answer.

To get the range, without using calculus (much easier), we have to think of what value of $x$ will give use the greatest value of $y$. Since the value of $- {\left(x - 2\right)}^{2}$ will always be negative, then 0 will be be the largest number we can get from any value of x. If we solve for this we get the following:
$x - 2 = 0$
$x = 2$

Plugging in $x = 2$ we get
$y \left(2\right) = - {\left(2 - 2\right)}^{2} + 3$
$= - {\left(0\right)}^{2} + 3$
$= 0 + 3$
$= 3$

The smallest value of y will then be $- \setminus \infty$ because the larger $x$ gets, the larger $- {\left(x - 2\right)}^{2}$ will be, but it will always be negative. Since 3 doesn't really have much of an impact on really big numbers, the range is $\left(- \setminus \infty , 3\right]$.