91Ó°ÊÓ

The domain and range are essential when understanding the behavior of a reciprocal function like \( g(x) = \frac{1}{x} \). To determine the domain, we need to find all the possible input values \( x \) can assume.
In this instance, the domain includes all real numbers except \( x = 0 \). That's because we can't divide by zero — it's undefined.
So, for any real number except zero, the function \( g(x) = \frac{1}{x} \) will yield a result.

Regarding the range, we focus on the possible outputs or values that \( g(x) \) can produce. For \( g(x) = \frac{1}{x} \), the range is also all real numbers except zero. This is because \( g(x) \) never actually hits zero. As \( x \) approaches zero, \( g(x) \) becomes increasingly large or small, moving towards infinity or negative infinity, but it never actually reaches zero. Thus, \( y eq 0 \) is the range.

Asymptotes play a crucial role in shaping the graph of a reciprocal function like \( g(x) = \frac{1}{x} \). They are lines that the graph approaches but never reaches. For \( g(x) \), there are two key asymptotes to consider.

The vertical asymptote occurs at \( x = 0 \). This vertical line symbolizes that no matter how close \( x \) gets to zero, \( g(x) \) will skyrocket to positive or negative infinity.
Since \( g(x) \) is undefined at \( x = 0 \), the graph will never touch this line.

Additionally, there's a horizontal asymptote at \( y = 0 \). This horizontal line signifies that as \( x \) becomes extremely large or small, \( g(x) \) nears the value of zero, but it never actually rests on the \( x \)-axis.
The function approaches zero, getting tinier and tinier in magnitude, but never quite reaching zero.

Graphing a reciprocal function like \( g(x) = \frac{1}{x} \) offers a visual representation of how the function behaves. It's critical to observe the two main parts of such a graph.

The graph of \( g(x) = \frac{1}{x} \) consists of two distinct, curved branches. One branch is located in the first quadrant, where both \( x \) and \( y \) are positive. Here, as \( x \) increases, \( g(x) \) decreases, approaching the \( x \)-axis but never touching it.
In the third quadrant, a similar mirrored branch exists where both \( x \) and \( y \) are negative.
Again, the function approaches the same horizontal asymptote of \( y = 0 \).

The graph of \( g(x) \) shows how values of \( x \) that draw closer to zero result in \( g(x) \) stretching towards infinity (positive above the horizontal asymptote and negative below).

• In the first quadrant, as \( x \) approaches zero from the positive side, \( g(x) \) heads to positive infinity.
• Conversely, in the third quadrant, as \( x \) comes from the negative side, \( g(x) \) dives to negative infinity.

This pattern reinforces the idea of the asymptotes and clearly defines the behavior and output of reciprocal functions.