What is the x -value of the hole? \( g(x)=\frac{(x-2)(x+6)}{(x+2)(x-2)} \)

To find the x-value of the hole in the function \( g(x)=\frac{(x-2)(x+6)}{(x+2)(x-2)} \), we need to look for the common factors in the numerator and denominator. Here, both the numerator and denominator have the factor \( (x-2) \). This gives us a hole at the x-value where \( x-2 = 0 \). Thus, solving for x: \( x - 2 = 0 \) \( x = 2 \) The hole in the graph of \( g(x) \) is located at \( x = 2 \). To further understand, it's fascinating to realize that holes in rational functions often occur at points where both the numerator and the denominator equal zero at the same time. This highlights the delicacy of certain points in mathematics – they remind us that not every number gets to be a star!