Find the Domain of y = 4x² - 49/100: Positive and Negative Regions

The function given is $y = 4x^2 - \frac{49}{100}$, and we need to analyze where it is positive and negative.

First, let's find the roots by setting the function equal to zero:

$4x^2 - \frac{49}{100} = 0$

Solve for $x$:

$4x^2 = \frac{49}{100}$

$x^2 = \frac{49}{400}$

$x = \pm \frac{7}{20}$

We have roots at $x = \frac{7}{20}$ and $x = -\frac{7}{20}$. These roots divide the real line into three intervals: $(-\infty, -\frac{7}{20})$, $(-\frac{7}{20}, \frac{7}{20})$, and $(\frac{7}{20}, \infty)$.

Since the coefficient of $x^2$ is positive (4), the parabola opens upwards, meaning the function is positive outside the interval between the roots and negative within it. Thus:

The function $y = 4x^2 - \frac{49}{100}$ is negative in the interval $-\frac{7}{20} < x < \frac{7}{20}$ and positive in the intervals $x < -\frac{7}{20}$ and $x > \frac{7}{20}$. Therefore:

The positive and negative domains are:

• Positive: $x < -\frac{7}{20}$ or $x > \frac{7}{20}$
• Negative: $-\frac{7}{20} < x < \frac{7}{20}$

Thus, the correct multiple-choice answer is:

$x < 0 : -\frac{7}{20} < x < \frac{7}{20}$

$x > \frac{7}{20}$ or $x > 0 : x < -\frac{7}{20}$