Problem 29 - College Algebra with Modeling and Visualization

Question

Find any horizontal or vertical asymptotes. $$ f(x)=\frac{3 x(x+2)}{(x+2)(x-1)} $$

Step-by-Step Solution

Answer

Vertical asymptote: \(x = 1\); Horizontal asymptote: \(y = 3\).

Step 1: Simplify the Function

To find asymptotes, you should start by simplifying the function if possible. The function given is \( f(x) = \frac{3x(x+2)}{(x+2)(x-1)} \).You can cancel the common factor \((x+2)\) in the numerator and the denominator, provided \(x eq -2\). Hence, the simplified function is:\[ f(x) = \frac{3x}{x-1} \] where \(x eq -2\).

Step 2: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator is nonzero at these points.For the simplified function \( f(x) = \frac{3x}{x-1} \), set the denominator equal to zero:\[ x - 1 = 0 \]Solve for \(x\):\[ x = 1 \]So, there is a vertical asymptote at \( x = 1 \). Since \(x = -2\) was a removable discontinuity, it is not a vertical asymptote.

Step 3: Determine Horizontal Asymptotes

Horizontal asymptotes occur based on the degrees of the polynomial in the numerator and the denominator.For \( f(x) = \frac{3x}{x-1} \), the degree of the numerator is 1 and the degree of the denominator is also 1.When the degrees are the same, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and the denominator respectively.Here, \( a = 3 \) and \( b = 1 \), so the horizontal asymptote is:\[ y = \frac{3}{1} = 3 \].

Key Concepts

Rational Functions

Rational functions are essential in mathematics and describe the ratio of two polynomials. This means that they are functions of the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \).
These functions can be extremely useful for representing real-world scenarios. For instance, they model relationships with non-linear growth or decay. These functions also include vertical and horizontal asymptotes, which significantly define their behavior.
One important aspect of rational functions is that they are not defined everywhere. The values that make \( Q(x) = 0 \), where the denominator is zero, are not included in the domain of the function.

Vertical Asymptotes

Vertical asymptotes in rational functions occur where the denominator equals zero, but the numerator does not. In simple terms, these are values that make the function's output approach infinity or negative infinity, thus creating a vertical line near these points.
To find a vertical asymptote, factor the denominator and solve for zero. In the example of the function \( f(x) = \frac{3x}{x-1} \), setting the denominator \( x-1 = 0 \) gives us a vertical asymptote at \( x = 1 \).
It's essential to differentiate between vertical asymptotes and removable discontinuities. A removable discontinuity, like the one at \( x = -2 \) previously mentioned, happens when a factor cancels out completely with one in the numerator.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as \( x \) approaches infinity or negative infinity. Essentially, they are horizontal lines that the graph of the function approaches as \( x \) gets larger or smaller in absolute value.
The rules to identify horizontal asymptotes depend on the degrees of the polynomials in the numerator and denominator:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, it is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator.

For the function \( f(x) = \frac{3x}{x-1} \), the degrees of both numerator and denominator are 1. Thus, the horizontal asymptote is \( y = \frac{3}{1} = 3 \).

Polynomial Division

Polynomial division is a mathematical process used to divide two polynomials, similar to how we perform arithmetic long division. This technique becomes handy when simplifying rational functions or finding asymptotes.
When you divide polynomials, you may end up simplifying the expression or finding where certain values cause the denominator to become zero, which is crucial for identifying vertical asymptotes and understanding the function's behavior.
For instance, simplifying \( f(x) = \frac{3x(x+2)}{(x+2)(x-1)} \) involves canceling out the common terms present in the numerator and denominator. This reduces to \( f(x) = \frac{3x}{x-1} \), making it easier to analyze for asymptotes.