Final answer:
The x-intercept of tan(x) within the interval (-π/2, π/2) is x = 0. This is because the x-intercept is the point where the output of the function is zero, and tan(x) is zero where the sinusoidal function crosses the x-axis, which within the given interval is only at zero.
Step-by-step explanation:
The question asks to find the x-intercept of the tan(x) function within the interval (-π/2, π/2). The x-intercept of a function is the point where the graph of the function crosses the x-axis, which corresponds to the output of the function being zero. Since the tangent function, tan(x), is the ratio of the sine to the cosine, the x-intercepts occur where the sine function is zero. The sine function is zero at multiples of π, therefore, within the specified interval, the x-intercept of tan(x) is at x = 0.
In this interval, the function tan(x) does not have any other x-intercepts because the sine function doesn't cross zero again within this interval. Therefore, the big final answer for the x-intercept of tan(x) in the interval (-π/2, π/2) is x = 0.
This is the solution while considering the periodic nature of the tangent function and the domain of the function in which we were asked to find the x-intercept. There are also nominal x-intercepts at every multiple of π, but those do not fall within the given interval, so they are not included in the big explanation of the solution for this particular case.