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

Feb 5, 2017

Domain: All $x \le - 2$ and $x \ge 7$
Range: All $y \ge 0$

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 $< 0$

${x}^{2} - 5 x - 14 < 0$ ...illegal values are negatives under roots
$\left(x + 2\right) \left(x - 7\right) < 0$ ...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 $x \le - 2$ and all $x > 7$

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

Range: All $y \ge 0$