When we delve into the function g(x) = 1/(x+1) + 1/(x+2), it's essential to understand its domain—where this mathematical expression is valid. The key lies in recognizing that division by zero is a no-go in mathematics. Therefore, we need to identify any values of x that would make either denominator equal to zero.
Starting with the first term, 1/(x+1), it becomes undefined when x + 1 = 0, which simplifies to x = -1. Moving on to the second term, 1/(x+2), it hits a snag when x + 2 = 0 or simply put, when x = -2.
This leads us to conclude that our function g(x) cannot take on these two specific values: -2 and -1. Thus, if we were plotting this function or analyzing its behavior across different intervals of real numbers (R), we'd have gaps at these points.
So what does this mean for the overall domain? We can express it as three distinct intervals: from negative infinity up until but not including -2; then from just above -2 through zero; and finally from zero extending out towards positive infinity. In mathematical notation, that's written as (-∞,-2) ∪ (-2,0) ∪ (0,+∞).
By breaking down where our function fails due to those pesky divisions by zero—and understanding how they shape our view—we get a clearer picture of where g(x) operates smoothly without interruption.