Problem 29 - Algebra and Trigonometry

Question

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each crational function. $$ f(x)=\frac{x^{2}-9}{x-3} $$

Step-by-Step Solution

Answer

The given function does not have any vertical asymptotes. However, there is a hole in the function's graph at x = 3.

Step 1: Simplify The Function

The given rational function is \(f(x)=\frac{x^{2}-9}{x-3}\). The numerator can be factored using the Difference of Squares formula, so we can simplify the function to: \(f(x)=\frac{(x-3)(x+3)}{x-3}\)

Step 2: Identify the Vertical Asymptotes and Holes

After cancellation, the simplified function is \(f(x) = x + 3\). If we put \(x-3 = 0\), we get \(x = 3\). However, since \(x = 3\) is a root of a common factor between the numerator and the denominator, this does not create an asymptote, but a hole in the graph. Hence, there are no vertical asymptotes for this function but there is a hole at x = 3.

Key Concepts

Holes in Rational Functions

In the study of rational functions, holes occur when there is a common factor in both the numerator and the denominator that can be cancelled out.
This simplified factor reveals a discontinuity at a certain point on the graph.
Unlike vertical asymptotes, which indicate that the function will zoom off towards positive or negative infinity at a certain x-value, holes are actual specific points on the graph where the function is undefined.

• Holes appear when a factor is cancelled (the factor exists in both the numerator and the denominator).
• To find a hole, solve the cancelled factor for zero.
• The x-value from this solution is where the hole occurs.

In the case of the function we studied, \(f(x) = \frac{x^2-9}{x-3}\), we simplified it to \(f(x) = x + 3\) by cancelling the \(x-3\) factor, revealing a hole at \(x = 3\).
When x equals 3, the original rational function becomes indeterminate, hence creating a hole. This means, on the graph of the function, there is a missing point at \(x = 3\), even though the line appears continuous.

Rational Functions

A rational function is any function that can be expressed as the ratio of two polynomials.
These functions can take on a variety of shapes and characteristics depending on the degrees and coefficients of the polynomials in the numerator and denominator.

• The general form is \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)eq 0\).
• Possible key features of rational functions include vertical asymptotes, holes, horizontal or slant asymptotes, and x-intercepts.
• The behavior of the function heavily relies on the relationship between the numerator and denominator.

In our exercise, we were tasked to analyze a rational function \(f(x) = \frac{x^2-9}{x-3}\). By understanding the role of numerator and denominator polynomials, we were able to simplify and interpret its graphical behavior effectively.

Difference of Squares

The difference of squares is a simple yet powerful algebraic technique used to simplify certain polynomial expressions.
It is based on the formula:
\[(a^2 - b^2) = (a - b)(a + b)\]
This identity allows us to factor expressions that match the form of a square number minus another square number, quickly and effectively.

• Perfect for simplifying quadratics that don't have a middle x-term.
• Helps in identifying cancellation opportunities in rational functions.
• Useful for finding holes in graphs of rational functions.

In our given function \(f(x) = \frac{x^2-9}{x-3}\), the numerator \(x^2 - 9\) is a difference of squares, as it can be expressed as \((x - 3)(x + 3)\).
This form makes it easier to identify and cancel common factors with the denominator, simplifying the rational expression and revealing potential holes at roots shared by both the numerator and the denominator.