Question

What is the domain and range of `sqrt((5x+6)/2)`?

Answer 1

Answer:

Domain `x in[-6/5,oo)`
Range `[0,oo)`

Explanation:

You must keep in mind that for the domain:

`sqrt(y)->y>=0`

`ln(y)->y>0`

`1/y->y!=0`

After that, you will be lead to an unequality giving you the domain.

This function is a combination of linear and square functions. Linear has domain `RR`. The square function though must have a positive number inside the square. Therefore:

`(5x+6)/2>=0`

Since 2 is positive:

`5x+6>=0`

`5x>= -6`

Since 5 is positive:

`x>= -6/5`

The domain of the functions is:

`x in[-6/5,oo)`

The range of the root function (outer function) is `[0,oo)` (infinite part can be proven through the limit as `x->oo`).