Question

What is the domain and range of `y =sqrt((x^2-5x-14))`?

Answer 1

Domain: All `x<=-2` and `x>=7`
Range: All `y>=0`

Explanation:

The domain can be described as all the "legal" values of `x`.

• You can't divide by zero
• You can't have negatives under a square root

If you find the "illegal" values, then you know the domain is all `x` except those!

The "illegal" values of `x` would be whenever the mantissa `< 0`

`x^2-5x-14<0` ...illegal values are negatives under roots
`(x+2)(x-7)<0` ...factor the left hand side

Now separate the two factors and flip one of the inequalities. One of the terms has to be negative (i.e., `<0`) and the other must be positive (i.e., `>0`).

`x+2 > 0` and `x-7 < 0`
`x > -2` and `x < 7`

The domain is all `x` except those illegal ones you just found.

Domain: All `x<=-2` and all `x>7`

The range are all values of `y` with domain `x`'s plugged in.

Range: All `y>=0`