Find the domain of g(x)=log(x+4x2−9)g(x)=log(x+4x2-9). Write the answer in interval notation.

To find the domain of the function \( g(x) = \log(x + 4x^2 - 9) \), we need to determine when the argument of the logarithm is positive:

\[
x + 4x^2 - 9 > 0
\]

This can be rearranged as:

\[
4x^2 + x - 9 > 0
\]

Next, we will find the roots of the quadratic equation \( 4x^2 + x - 9 = 0 \) using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 4 \), \( b = 1 \), and \( c = -9 \):

\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4}
\]
\[
x = \frac{-1 \pm \sqrt{1 + 144}}{8}
\]
\[
x = \frac{-1 \pm \sqrt{145}}{8}
\]

Calculating the roots:

\[
x_1 = \frac{-1 + \sqrt{145}}{8}, \quad x_2 = \frac{-1 - \sqrt{145}}{8}
\]

Now, let's evaluate the approximate values of the roots:

- \( \sqrt{145} \approx 12.04 \)
- Thus, \( x_1 \approx \frac{11.04}{8} \approx 1.38 \)
- And \( x_2 \approx \frac{-13.04}{8} \approx -1.63 \)

Now we have two roots: \( x_1 \approx 1.38 \) and \( x_2 \approx -1.63 \).

Next, we need to test the intervals defined by these roots: \( (-\infty, x_2) \), \( (x_2, x_1) \), and \( (x_1, \infty) \).

1. **Interval \( (-\infty, -1.63) \)**: Choose \( x = -2 \):
   \[
   4(-2)^2 + (-2) - 9 = 16 - 2 - 9 = 5 > 0
   \]

2. **Interval \( (-1.63, 1.38) \)**: Choose \( x = 0 \):
   \[
   4(0)^2 + (0) - 9 = -9 < 0
   \]

3. **Interval \( (1.38, \infty) \)**: Choose \( x = 2 \):
   \[
   4(2)^2 + (2) - 9 = 16 + 2 - 9 = 9 > 0
   \]

From this analysis, we find that the function is positive in the intervals:

- \( (-\infty, -1.63) \)
- \( (1.38, \infty) \)

Thus, the domain of \( g(x) \) in interval notation is:

\[
(-\infty, \frac{-1 - \sqrt{145}}{8}) \cup (\frac{-1 + \sqrt{145}}{8}, \infty)
\]

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. What is the range of the function \( g(x) \)?
2. How do you find the critical points of \( g(x) \)?
3. What is the behavior of \( g(x) \) at the boundaries of the domain?
4. How can you graph the function \( g(x) \)?
5. What happens to \( g(x) \) as \( x \) approaches the roots? 

**Tip**: Always check if the argument of a logarithmic function is positive to determine its domain!

Find the domain of g(x)=log(x+4x2−9). Write the answer in interval notation.

Explore how to determine the domain of the function g(x) = log(x + 4x² - 9) through step-by-step analysis of quadratic inequalities. The solution reveals the domain in interval notation: (-∞, (−1 - √145)/8) ∪ ((−1 + √145)/8, ∞). Suitable for high school students.

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