(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 fand $g$, and (e) explain why the differences in the domains of $f$ and $g$ are not shown in their graphs. $$f(x)=\frac{x^{2}-9}{x-3}, \quad g(x)=x+3$$

Key Concepts

Graphical Representation vs. Domain Restrictions

Graphical representations of functions may not always show all domain restrictions, especially removable discontinuities. When a rational function is simplified, common factors that cause holes are no longer visible in the graph, so while the graph appears continuous, the original function’s domain still excludes the values causing the original discontinuities.

Vertical Asymptotes

Vertical asymptotes are values of x for which the function's value approaches infinity or negative infinity, typically occurring when the denominator is zero and no factor cancellation occurs. They represent boundaries where the behavior of the function changes dramatically.

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 important to identify and exclude values that cause a zero in the denominator, as these values result in undefined expressions.

Rational Functions

Rational functions are expressions that represent the quotient of two polynomials. Analyzing these functions typically involves factoring and simplifying, which can reveal important characteristics such as discontinuities that are not always apparent from the simplified form.

Removable Discontinuities (Holes)

A removable discontinuity, or hole, occurs in the graph of a rational function when a factor cancels in the process of simplification. Despite the cancellation, the original function is still undefined at that particular x-value, resulting in a 'hole' in the graph.

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