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Asymptotes are lines that a graph approaches but never actually touches. In our case with the function \( f(x) = \frac{1}{x-5} \), we see both vertical and horizontal asymptotes.

Vertical asymptotes occur where the function is undefined. When you plug in \( x = 5 \), the function becomes undefined because the denominator equals zero. Thus, \( x = 5 \) is our vertical asymptote. It acts as a boundary the graph cannot cross, and as \( x \) gets infinitesimally close to 5, \( f(x) \) shoots off to positive or negative infinity.

Horizontal asymptotes give us behavior at the extremes as \( x \) approaches infinity. As \( x \) becomes extremely large or small, \( f(x) \) will get closer and closer to 0, making \( y = 0 \) the horizontal asymptote. This tells us the direction in which the branches of the hyperbola flatten out but never actually reach a y-value of zero.

Intercepts are points where the graph crosses the axes.

The y-intercept is found by setting \( x \) to 0, leading us to calculate \( f(0) = \frac{1}{-5} = -\frac{1}{5} \). This gives us the y-intercept at the point \((0, -\frac{1}{5})\), meaning the graph crosses the y-axis there. It's a peculiar point because often you'll see the graph hitting somewhere at a more standard axis location, but it reflects how every graph is unique in its own right.

The x-intercept is the point where \( f(x) \) equals zero. However, for this function, you will find there are no x-intercepts. An x-intercept would require the numerator, which is always 1 here, to be zero. Since 1 is never zero, our function never actually touches the x-axis, perfectly aligning with the horizontal asymptote.

Functions map input values to output values following a specific rule. For our function \( f(x) = \frac{1}{x - 5} \), the relationship involves division by \( (x - 5) \).

Understanding a function's domain and range is key to grasping its behavior. The domain refers to all x-values making the function valid. Here, any real number except \( x = 5 \) is part of the domain. The exclusion stems from division by zero, which is undefined in mathematics.

The range, on the other hand, consists of all possible y-values. Since \( f(x) \) can approach any real number but gets infinitely large or small around \( x = 5 \) and never reaches zero, every real number is part of the range except \( y = 0 \). The study of this function lets us visualize how especially with undefined points the graph steers clear horizontally and vertically, creating those unique asymptotes.