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An asymptote is a line that a graph approaches but never touches or crosses. For our function, we identified a vertical asymptote.

The vertical asymptote is found where the function becomes undefined. For the function \(f(x) = \frac{x^2 - 9}{x + 3}\), the denominator becomes zero at \(x = -3\). After simplifying by cancelling \(x + 3\), we see that \(f(x)\) is undefined at \(x = -3\), indicating a vertical asymptote at that point.

It's important to note that vertical asymptotes are distinct from holes in the graph. A hole occurs if both the numerator and denominator can be cancelled out at the undefined point. In our function, we cancel the \(x + 3\) factor, but the simplification still leaves us with an undefined point at \(x = -3\). Hence, this is a vertical asymptote, not a hole. Always graph these carefully, sometimes as dotted lines, to represent their presence.

The domain of a function includes all possible values of \(x\) for which the function is defined.

In our function \(\frac{x^2 - 9}{x + 3}\), it simplifies to \(x - 3\) but is still undefined at \(x = -3\). Thus, excluding this value from the domain is crucial.

The domain is therefore all real numbers except \(-3\). Written in set notation, the domain is: \[ \{ x \in \mathbb{R} \mid x eq -3 \} \]

The range of the function represents all possible output values. After simplification, our function becomes \(f(x) = x - 3\). Since this is a linear function with a slope of 1, its range is all real numbers \(y \in \mathbb{R}\).

Understanding how to find domain and range helps predict the behavior of the function across its entire graph.

Intercepts are points where the graph of a function crosses the axes.

To find the \(x\)-intercept (where the graph crosses the x-axis), set \(f(x) = 0\). With our simplified function \(f(x) = x - 3\), solving \(0 = x - 3\) gives \(x = 3\). So, the \(x\)-intercept is at \((3, 0)\).

Finding the \(y\)-intercept involves setting \(x = 0\). Substituting 0 into the function, \(f(0) = 0 - 3 = -3\). Thus, the \(y\)-intercept is at \((0, -3)\).

Correctly identifying intercepts is crucial, as these points provide valuable insight into the shape and position of the graph.