Question

How do you find the domain and range of `sqrt(x^2-5x-14)`?

Answer 1

`(-oo, -2] U [7, oo)`

Explanation:

Domain is the possible values for `x`.

The domain for the parent function , `sqrtx` is from `0` to `oo`, or `[0, oo)`
graph{y=sqrtx}

Notice how `0` is included. That's because `sqrt0` is the lowest number that is possible. Any number smaller than `0` is negative, and a square root of a negative number craetes an unreal number (`i`), which we can't graph.

Long story short, we can't take the square rot of a negative number, so we stop at `0`

Let's factor the quadratic and see the roots for `x`:

`x^2-5x-14`

We need to find `2` numbers that multiply to `-14` and add to `-5`

`color(white)(xx) + -5`
`color(white)(+) xx -14`
.........................
`+-1 xx +-14`
`color(red)(+-2 xx +-7)`

`-7+2` gives us `-5`, and `-7 xx 2` equals `-14`

So, now we have `sqrt((x+2)(x-7))`

So, the rule with square roots is that the factors must be equal to or larger than `0`

Let's solve for these roots and see what value of `x` makes everything equal `0`

factor 1
`x+2=0`

`x=-2`

factor 2
`x-7=0`

`x=7`

Now we know that if `x` equals `-2` or `7`, we're taking a square root of `0`. So, if `x` equals anything smaller than `7` or larger than `-2`, we would be taking a square root of a negative number.

So, our domain is "anything smaller than `-2` and anything larger than `7`", or `(-oo, -2] U [7, oo)`

To check our work, let's graph the equation:
graph{y=sqrt((x+2)(x-7))}

Yep! We were right.

The graph has no issues until `-2` to `7`. From those points, the graph is not possible