(a) determine the domains of f and g, (b) find any vertical...

(a) determine the domains of $f$ and $g,$ (b) find any vertical asymptotes and holes in the graphs of $f$ and $g,$ (c) compare $f$ and $g$ by completing the table, (d) use a graphing utility to graph $f$ and $g,$ and (e) explain why the differences in the domains of $f$ and $g$ are not shown in their graphs. $$f(x)=\frac{x^{2}-9}{x-3}, \quad g(x)=x+3$$ $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline f(x) & & & & & & & \\ \hline g(x) & & & & & & & \\ \hline \end{array}$$

(a) determine the domains of $f$ and $g,$
(b) find any vertical asymptotes and holes in the graphs of $f$ and $g,$
(c) compare $f$ and $g$ by completing the table,
(d) use a graphing utility to graph $f$ and $g,$ and
(e) explain why the differences in the domains of $f$ and $g$ are not shown in their graphs.
$$f(x)=\frac{x^{2}-9}{x-3}, \quad g(x)=x+3$$
$$\begin{array}{|l|l|l|l|l|l|l|l|}
\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline f(x) & & & & & & & \\
\hline g(x) & & & & & & & \\
\hline
\end{array}$$

Key Concepts

Graphing Functions with Discontinuities

When graphing functions, especially those involving rational expressions, it is important to note that graphing utilities may display the simplified form of the function, which can obscure discontinuities such as holes. Understanding the algebraic processes behind simplification helps explain discrepancies between the algebraic domain and the graph's appearance.

Removable Discontinuities (Holes)

A removable discontinuity, or a hole, occurs when a factor in the numerator and denominator cancels out, leading to an undefined point in the original function despite the simplified version appearing continuous. These points are often not immediately evident in the graph since the function behaves like the simplified expression everywhere else.

Rational Functions and Factorization

Rational functions are defined as the ratio of two polynomials. Factorization is a key technique used to simplify these functions by breaking down the numerator and the denominator into their factors. This process can reveal common factors that may cancel out, often leading to a simplified form of the function.

Domain of a Function

The domain of a function is the complete set of input values for which the function is defined. In the context of rational functions, it is particularly important to identify values that would make any denominator equal to zero, as these values are excluded from the domain.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at the values of the variable which make the denominator equal to zero while leaving the numerator non-zero. They represent lines that the graph of the function approaches but never actually touches, indicating infinite behavior near these points.

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