# How do you find the domain and range of  sqrt(25-(x-2)^2) +3?

Jan 30, 2017

The domain is $x \in \left[- 3 , 7\right]$
The range is $y \in \left[3 , 8\right]$

#### Explanation:

Let $g \left(x\right) = \sqrt{25 - {\left(x - 2\right)}^{2}} + 3$

What is under the sqrt sign is $\ge 0$. this is the domain

So,

$25 - {\left(x - 2\right)}^{2} \ge 0$

$25 - \left({x}^{2} - 4 x + 4\right) \ge 0$

${x}^{2} - 4 x + 4 - 25 \le 0$

${x}^{2} - 4 x - 21 \le 0$

Let's factorise

$\left(x - 7\right) \left(x + 3\right) \le 0$

Let $f \left(x\right) = \left(x - 7\right) \left(x + 3\right)$

Let 's do a sign chart to solve this inequality

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 3$$\textcolor{w h i t e}{a a a a}$$7$$\textcolor{w h i t e}{a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 3$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$x - 7$$\textcolor{w h i t e}{a a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$

Therefore,

$f \left(x\right) \le 0$ when $x \in \left[- 3 , 7\right]$, this is the domain

To calculate the range,

When $x = - 3$, $\implies$, $g \left(- 3\right) = 3$

When $x = 7$, $\implies$, $g \left(7\right) = 3$

When $x = 2$, $\implies$, $g \left(2\right) = 8$

Let $y = \sqrt{25 - {\left(x - 2\right)}^{2}} + 3$

The range is $y \in \left[3 , 8\right]$

graph{(sqrt(25-(x-2)^2)+3) [-9.74, 12.76, -2.055, 9.195]}