Question

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

Answer 1

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

Explanation:

Let `g(x)=sqrt(25-(x-2)^2)+3`

What is under the `sqrt` sign is `>=0`. this is the domain

So,

`25-(x-2)^2>=0`

`25-(x^2-4x+4)>=0`

`x^2-4x+4-25<=0`

`x^2-4x-21<=0`

Let's factorise

`(x-7)(x+3)<=0`

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

Let 's do a sign chart to solve this inequality

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaa)` `7` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `x-7` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

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

To calculate the range,

When `x=-3`, `=>`, `g(-3)=3`

When `x=7`, `=>`, `g(7)=3`

When `x=2`, `=>`, `g(2)=8`

Let `y=sqrt(25-(x-2)^2)+3`

The range is `y in [3,8]`

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