To find the [tex]$x$[/tex]-intercepts of the graph of the function [tex]\( y = 6 \tan \left( \frac{x}{2} \right) - 3 \)[/tex], we need to determine where the function crosses the [tex]\( x \)[/tex]-axis. In other words, we need to solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex].
1. Start with the equation:
[tex]\[
6 \tan \left( \frac{x}{2} \right) - 3 = 0
\][/tex]
2. Isolate the tangent function by adding 3 to both sides:
[tex]\[
6 \tan \left( \frac{x}{2} \right) = 3
\][/tex]
3. Divide both sides by 6:
[tex]\[
\tan \left( \frac{x}{2} \right) = \frac{1}{2}
\][/tex]
4. To solve for [tex]\( x \)[/tex], consider the equation [tex]\( \tan \left( \frac{x}{2} \right) = \frac{1}{2} \)[/tex]. The tangent function has a period of [tex]\( \pi \)[/tex], so the general solution is:
[tex]\[
\frac{x}{2} = \arctan \left( \frac{1}{2} \right) + n\pi \quad \text{for any integer } n
\][/tex]
5. Isolate [tex]\( x \)[/tex] by multiplying both sides by 2:
[tex]\[
x = 2 \arctan \left( \frac{1}{2} \right) + 2n\pi
\][/tex]
6. The principal value of [tex]\( \arctan \left( \frac{1}{2} \right) \)[/tex] is approximately [tex]\( 0.4636 \)[/tex]. Therefore:
[tex]\[
2 \arctan \left( \frac{1}{2} \right) \approx 2 \times 0.4636 \approx 0.9273
\][/tex]
7. Hence, the [tex]\( x \)[/tex]-intercepts of the function are:
[tex]\[
x = 0.9273 + 2n\pi \quad \text{for any integer } n
\][/tex]
So, the correct form of the [tex]\( x \)[/tex]-intercepts is:
[tex]\[
(0.9273 + 2n \pi, 0) \quad \text{where } n \text{ is any integer}
\][/tex]
Thus, the correct answer is:
[tex]\[
(0.9273 + 2n \pi, 0) \quad \text{where } n \text{ is any integer}
\][/tex]
