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The domain of a function refers to the set of all possible input values (typically represented by \( x \)) that will produce a valid output. In simpler terms, it's the range of \( x \)-values that you can plug into the function without breaking any mathematical rules, like dividing by zero. For the function \( f(x) = \frac{x^2 - 4}{x - 2} \), you need to find which \( x \)-values make the denominator zero because division by zero is undefined.

To find the domain of \( f(x) \):

• Set the denominator equal to zero to find any restrictions: \( x - 2 = 0 \).
• Solve this equation, which gives \( x = 2 \).
• This means that \( x = 2 \) is not included in the domain.

Thus, the domain of \( f(x) \) is all real numbers except \( x = 2 \). In interval notation, this is written as \( (-\infty, 2) \cup (2, \infty) \). By understanding the domain, you ensure you're working within the "safe zone" of the function where it operates correctly.

Graphing rational functions involves plotting the function on a coordinate plane to visualize its behavior and see where any points of discontinuity or holes occur. For the function \( f(x) = \frac{x^2 - 4}{x-2} \), you want to simplify first, if possible. Noticing the numerator can be factored as a difference of squares \((x-2)(x+2)\), the function simplifies to \( f(x) = x + 2 \) for all \( x eq 2 \).

When graphing:

• Draw the simplified expression \( y = x + 2 \), a straight line.
• Remember that at \( x = 2 \), the original function is undefined leading to a "hole" on the graph. Indicate this point with an open dot.
• Choose an appropriate range in the viewing rectangle, particularly surrounding \( x = 2 \), to clearly show this discontinuity.

By graphing, you're able to see the behavior of the function over its domain and spot where it doesn't work properly.

Asymptotes in a graph are lines that the graph approaches but never actually reaches. They're crucial in understanding how a function behaves at its extremes. For rational functions, horizontal asymptotes can be found by examining the behavior of the function as \( x \) approaches infinity or negative infinity, while vertical asymptotes are found where the function blows up because of division by zero.

In the case of \( f(x) = x + 2 \), which comes from simplifying \( \frac{x^2 - 4}{x-2} \):

• There is no horizontal asymptote because the simplified function \( x + 2 \) is a linear line without a horizontal limit as \( x \) increases or decreases indefinitely.
• No vertical asymptotes exist because the discontinuity at \( x = 2 \) is a removable one, evident from the factor \( (x-2) \) that cancels out.

While in this particular function there are no asymptotes, understanding their importance in other rational functions helps predict how the graph behaves as \( x \) moves towards extremes.