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

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

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Answer 1 · 2017-01-30T11:11:26

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

Explanation:

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

What is under the $\sqrt{}$ $sqrt$ sign is $\geq 0$ $>=0$ . this is the domain

So,

$25 - {(x - 2)}^{2} \geq 0$ $25-(x-2)^2>=0$

$25 - (x^{2} - 4 x + 4) \geq 0$ $25-(x^2-4x+4)>=0$

$x^{2} - 4 x + 4 - 25 \leq 0$ $x^2-4x+4-25<=0$

$x^{2} - 4 x - 21 \leq 0$ $x^2-4x-21<=0$

Let's factorise

$(x - 7) (x + 3) \leq 0$ $(x-7)(x+3)<=0$

Let $f (x) = (x - 7) (x + 3)$ $f(x)=(x-7)(x+3)$

Let 's do a sign chart to solve this inequality

$a a a a$ $color(white)(aaaa)$ $x$ $x$ $a a a a$ $color(white)(aaaa)$ $- \infty$ $-oo$ $a a a a$ $color(white)(aaaa)$ $- 3$ $-3$ $a a a a$ $color(white)(aaaa)$ $7$ $7$ $a a a a$ $color(white)(aaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $x + 3$ $x+3$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $-$ $-$

$a a a a$ $color(white)(aaaa)$ $x - 7$ $x-7$ $a a a a a$ $color(white)(aaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $-$ $-$

$a a a a$ $color(white)(aaaa)$ $f (x)$ $f(x)$ $a a a a a a$ $color(white)(aaaaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $+$ $+$

Therefore,

$f (x) \leq 0$ $f(x)<=0$ when $x \in [- 3, 7]$ $x in [-3,7]$ , this is the domain

To calculate the range,

When $x = - 3$ $x=-3$ , $\Rightarrow$ $=>$ , $g (- 3) = 3$ $g(-3)=3$

When $x = 7$ $x=7$ , $\Rightarrow$ $=>$ , $g (7) = 3$ $g(7)=3$

When $x = 2$ $x=2$ , $\Rightarrow$ $=>$ , $g (2) = 8$ $g(2)=8$

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

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

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

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

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

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