Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

(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. $$f(x)=\frac{x^{2}-9}{x-3}, \quad g(x)=x+3$$ $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline f(x) & & & & & & & \\ \hline g(x) & & & & & & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The domain of \(f(x)\) is all real numbers except 3, and the domain of \(g(x)\) is all real numbers. \(f(x)\) has a hole in its graph at \(x=3\), and \(g(x)\) has no holes or asymptotes. Except for \(x=3\), \(f(x)\) and \(g(x)\) share the same values. The graph of both functions will look the same, except that \(f(x)\) will have a hole at \(x=3\). Even though \(f(x)\) has this hole, it doesn't appear in the graph because it's a single point, which is typically not visible in the graph.

Step 1: Determining the Domains

For function \(f(x)\), an undefined value for \(x=3\) exists because the denominator becomes 0, so the domain for \(f(x)\) is all real numbers except 3. The function \(g(x)\) is a linear function, therefore the domain for \(g(x)\) is all real numbers.

Step 2: Finding Asymptotes and Holes

We can factorize \(f(x)\) as follows: \(f(x)=\frac{(x-3)(x+3)}{x-3}\), if \(x≠3\). Thus, the function can be simplified as \(f(x)=x+3\) if \(x≠3\). The function \(f(x)\) has a hole at \(x=3\) as the value at \(x=3\) is undefined. For \(g(x)\), as a linear function, it has no asymptotes or holes.

Step 3: Completing the Table

Fill the table with values obtained from evaluating functions \(f(x)\) and \(g(x)\) at given \(x\). Whenever \(x≠3\), both functions have the same value. However, \(f(x)\) has an undefined value at \(x=3\).

Step 4: Graphing the Functions

Both functions will have identical graphs except at \(x=3\). There will be a hole in the graph of \(f(x)\) at \(x=3\). The graph of \(g(x)\) is a straight line with slope of 1 and y-intercept of 3.

Step 5: Explaining the Differences in the Domains

Despite the fact that the domains of \(f(x)\) and \(g(x)\) are distinct, the differences do not appear in their graphs. This is due to the hole in \(f(x)\) graph at \(x=3\). That single point doesn't show because we usually can't see a single point missing from a graph.

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (usually represented as \(x\)) for which the function is defined. For the function \(f(x) = \frac{x^2 - 9}{x - 3}\), there is a specific restriction: you cannot divide by zero. So, when \(x = 3\), the denominator becomes zero, making \(f(x)\) undefined at this point. Thus, the domain for \(f(x)\) is all real numbers except \(x = 3\).

• So, in interval notation, the domain of \(f(x)\) is \((-\infty, 3) \cup (3, \infty)\).

On the other hand, \(g(x) = x + 3\) is a linear function. Linear functions are very simple: for every \(x\), there is a corresponding \(g(x)\). This means there are no restrictions, and \(g(x)\) is defined for all real numbers.

• The domain of \(g(x)\) is \((-\infty, \infty)\).

Vertical Asymptotes

Vertical asymptotes are lines that a graph approaches but never actually touches or crosses. They occur in rational functions when the denominator is zero, causing the function to become undefined. For \(f(x)\), once we simplify \(\frac{(x - 3)(x + 3)}{x - 3}\) to \(x + 3\) for \(x eq 3\), we see there is no vertical asymptote since we removed the factor making the denominator zero.
However, this leaves a hole, which we'll explore next.

• Remember, \(f(x)\) does not have a vertical asymptote because the problematic factor cancels out.

In contrast, the function \(g(x) = x + 3\) is linear and doesn't involve any division by zero. Thus, linear functions like \(g(x)\) do not have vertical asymptotes.

Holes in Graphs

Holes in graphs occur where a function is not defined due to a cancelled factor in the numerator and denominator. For the function \(f(x)\), the expression simplifies to \(f(x) = x + 3\) for \(x eq 3\). This means there is a hole in the graph at \(x = 3\).

Why is this? Initially, the denominator being zero at \(x = 3\) would suggest a vertical asymptote, but because the factor \((x - 3)\) cancels out, the graph cannot "climb" up or down to infinity there. Instead, there's a gap—a point where \(f(x)\) is undefined.

• Such holes are often depicted as an open circle on the graph.

For \(g(x) = x + 3\), there are no such cancellations, so the graph is uninterrupted, possessing neither holes nor vertical asymptotes.

Graphing Utility

A graphing utility, like a calculator or software, is a tool used to plot the graph of functions. It's particularly useful in visualizing functions like \(f(x)\) and \(g(x)\). When graphing these functions:

• Plot \(g(x) = x + 3\) as a straight line with a constant slope of 1 and a y-intercept at 3.
• \(f(x)\), when graphed, will look identical to \(g(x)\) except for the hole at \(x = 3\).

Graphing utilities often cannot explicitly show a single point of discontinuity, such as a hole. However, by analyzing the algebraic form, you can know where this point occurs. This knowledge helps in situations where exact continuity is crucial, such as engineering tasks or data modeling.

Linear Functions

Linear functions are the simplest types of functions, represented by straight lines on a graph. They follow the standard form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. With \(g(x) = x + 3\), it is clear that:

• The slope \(m = 1\) indicates the line rises one unit for each unit increase in \(x\).
• The y-intercept \(b = 3\) shows where the line crosses the y-axis.

Linear functions, because they have no denominators (like fractions), naturally avoid having vertical asymptotes or holes. This makes them especially reliable for predicting outcomes or interpreting data where smooth continuity is paramount.
In contrast, rational functions, like \(f(x)\), can be more complex due to the potential for undefined points or asymptotic behavior.