Problem 83 - Algebra and Trigonometry

Question

In this chapter we adopted the convention that in rational functions, the numerator and denominator don't share a common factor. In this exercise we consider the graph of a rational function that doesn't satisfy this rule. (a) Show that the graph of $$r(x)=\frac{3 x^{2}-3 x-6}{x-2}$$ is the line \(y=3 x+3\) with the point \((2,9)\) removed. [Hint: Factor. What is the domain of \(r\)?] (b) Graph the rational functions: $$\begin{aligned} s(x) &=\frac{x^{2}+x-20}{x+5} \\ t(x) &=\frac{2 x^{2}-x-1}{x-1} \\ u(x) &=\frac{x-2}{x^{2}-2 x} \end{aligned}$$

Step-by-Step Solution

Answer

The function \( r(x) \) is the line \( y = 3x + 3 \) with a hole at point \((2, 9)\).

Step 1: Factor the numerator

The function given is \( r(x) = \frac{3x^2 - 3x - 6}{x - 2} \). Start by factoring the numerator \( 3x^2 - 3x - 6 \). We can factor this expression as follows: \( 3(x^2 - x - 2) \). The quadratic \( x^2 - x - 2 \) can be factored into \((x - 2)(x + 1)\). Thus, the factored numerator becomes \( 3(x - 2)(x + 1) \).

Step 2: Cancel common factors

After factoring the numerator and rewriting it as \( 3(x - 2)(x + 1) \), we see that the \( x - 2 \) term is common in both the numerator and the denominator. Cancel these terms to simplify the function to \( r(x) = 3(x + 1) \).

Step 3: Identify the line equation and domain

The resulting function after cancellation is \( r(x) = 3(x + 1) = 3x + 3 \), which is the equation of a straight line. However, we cancelled \( x - 2 \) from the denominator, which means \( x = 2 \) is not part of the domain, making it a hole in the graph at \((2, 9)\).

Step 4: Graph the rational functions

To graph, start with \( s(x) = \frac{x^2 + x - 20}{x + 5} \). Factor the numerator \( (x - 4)(x + 5) \). After cancelling \( x+5 \), it's \( s(x) = x - 4 \) with a hole at \( x = -5 \). For \( t(x) = \frac{2x^2 - x - 1}{x - 1} \), factor the numerator \( (2x + 1)(x - 1) \); after cancellation, it's \( t(x) = 2x + 1 \) with a hole at \( x = 1 \). \( u(x) = \frac{x-2}{x^2-2x} \) simplifies to \( \frac{1}{x} \) by cancelling \( x-2 \), creating holes at \( x=0 \) and \( x=2 \).

Step 5: Graph interpretation

Each rational function represents a line with points removed due to the cancelled factors. \( r(x) = 3x + 3 \) with a hole at \( (2, 9) \). \( s(x) = x - 4 \) with a hole at \( (-5, -9) \). \( t(x) = 2x + 1 \) with a hole at \( (1, 3) \). \( u(x) = \frac{1}{x} \) behaves like a hyperbola with holes at \( x = 0 \) and \( x = 2 \).

Key Concepts

Factoring

Factoring is a key skill that helps simplify rational functions by breaking down complex expressions into simpler ones. When we factor, we look for common elements in a mathematical expression that can be simplified or canceled out.
To factor a polynomial, like the numerator in the example function \( r(x) = \frac{3x^2 - 3x - 6}{x - 2} \), we begin by determining the greatest common factor, which in this case is 3. This gives us \( 3(x^2 - x - 2) \). Next, we factor the quadratic \( x^2 - x - 2 \) into \((x-2)(x+1)\). The complete factored form of the numerator becomes \( 3(x-2)(x+1) \).
This process of factoring is crucial as it allows us to simplify the function by identifying terms that can be canceled with those in the denominator. In rational functions, if both the numerator and the denominator share a factor, they can be simplified further. This leads to a clearer understanding of the function's behavior.

Graphing Rational Functions

Graphing rational functions involves translating the algebraic expression into a visual representation on a coordinate plane. This helps us understand the behavior of the function, including its asymptotes and intercepts.
When graphing a rational function, start by simplifying it if possible. In our main example, \( r(x) = \frac{3x^2 - 3x - 6}{x - 2} \) simplifies to \( 3(x + 1) \) after canceling the \( x-2 \) from both numerator and denominator. The graph of \( r(x) \) becomes the line \( y = 3x + 3 \).
Despite the simplification, the graph maintains certain characteristics from the original function. For instance, any restrictions on the domain or points where factors were canceled result in holes or gaps at specific points on the graph. These need to be clearly marked when graphing. Using graphing software or plotting points manually on paper provides visual confirmation of the algebraic interpretation.

Domain and Holes in Graphs

The domain of a rational function is all the real values that \( x \) can take. However, there are restrictions due to the nature of division by zero. These restrictions occur at values of \( x \) that make the denominator zero.
In our example, the function \( r(x) = \frac{3x^2 - 3x - 6}{x - 2} \) has its denominator \( x-2 \) equal to zero when \( x = 2 \). Therefore, \( x = 2 \) is not included in the domain of the function. In graphical terms, this omission is represented as a hole at \( (2, 9) \).
This hole indicates that while the line \( y = 3x + 3 \) can be extended infinitely, it will skip this particular point. Recognizing and representing holes in graphs is vital as they are not visible directly in the simplified equation. Always remember to mark these discontinuities to accurately reflect the function's character.