Find the Domain of (x+8)²-2¼: Complete Function Analysis

Find the positive and negative domains of the function below:

$y=\left(x+8\right)^2-2\frac{1}{4}$

The given function is $y = (x+8)^2 - 2\frac{1}{4}$. To find where it is positive or negative, we first find the points where $y = 0$.

Set the function equal to zero:

$(x+8)^2 - \frac{9}{4} = 0$

To handle the fraction, rewrite the equation:

$(x+8)^2 = \frac{9}{4}$

Now, take the square root of both sides:

$x+8 = \pm \frac{3}{2}$

This gives us two solutions for $x$:

$x = -8 + \frac{3}{2}$ and $x = -8 - \frac{3}{2}$

Calculating these values, we have:

$x = -\frac{16}{2} + \frac{3}{2} = -\frac{13}{2} = -6\frac{1}{2}$

$x = -\frac{16}{2} - \frac{3}{2} = -\frac{19}{2} = -9\frac{1}{2}$

Next, test intervals around the roots to determine where the function is positive or negative:

• For $x < -9\frac{1}{2}$, an example point is $x = -10$: $(-10+8)^2 - \frac{9}{4} = 4 - \frac{9}{4}$, which is positive.
• For $-9\frac{1}{2} < x < -6\frac{1}{2}$, an example point is $x = -8$: $(-8+8)^2 - \frac{9}{4} = 0 - \frac{9}{4}$, which is negative.
• For $x > -6\frac{1}{2}$, an example point is $x = -6$: $(-6+8)^2 - \frac{9}{4} = 4 - \frac{9}{4}$, which is positive.

This leads to the conclusion that:

$x < 0 :-9\frac{1}{2} < x < -6\frac{1}{2}$

And for $x > 0$, we have $x > -6\frac{1}{2}$ or $x < 0 : x < -9\frac{1}{2}$

Thus, the correct answer is:

$x < 0 :-9\frac{1}{2} < x < -6\frac{1}{2}$

$x > -6\frac{1}{2}$ or $x > 0 : x < -9\frac{1}{2}$