Problem 55 - Algebra 2

Question

Determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function. \(f(x)=\frac{x^{2}+4 x+3}{x+3}\)

Step-by-Step Solution

Answer

The graph has a hole at \(x = -3\); no vertical asymptotes exist.

Step 1: Simplify the Function

The function given is \(f(x)=\frac{x^{2}+4x+3}{x+3}\). First, we need to factor the numerator. \(x^{2}+4x+3\) can be factored into \((x+3)(x+1)\). Thus, the function simplifies to \(f(x) = \frac{(x+3)(x+1)}{x+3}\).

Step 2: Identify and Cancel Common Factors

Notice that \(x+3\) is a common factor in both the numerator and denominator. Cancel the common factor \(x+3\) from the function. The simplified function is now \(f(x) = x+1\), but the original division introduces restrictions.

Step 3: Determine the Vertical Asymptotes

Vertical asymptotes occur when the denominator of the original function is zero and there is no common factor in the numerator at that zero. After cancellation, there are no common factors in the numerator at \(x = -3\), pointing to a hole instead of a vertical asymptote.

Step 4: Identify Holes in the Graph

A hole occurs at a point where the function is not defined after canceling a common factor. Since \(x+3\) was canceled, the function has a hole at the value \(x = -3\).

Key Concepts

Rational Functions

Rational functions are a type of function that takes the form of a fraction where both the numerator and the denominator are polynomials. They can be written as \[ f(x) = \frac{P(x)}{Q(x)} \]where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)eq 0\). These functions are interesting because they can exhibit behavior like asymptotes, holes, or be simply defined over many values of \(x\). Understanding the behavior of the polynomial parts can help predict where the function is undefined or infinitely large. To analyze these, students often factor and simplify both parts of the function.
Often, this involves finding the zeros of the denominator, which helps identify vertical asymptotes or discontinuities such as holes in the graph.

Factoring Polynomials

Factoring polynomials is like breaking them into simpler multiplication parts. It's particularly useful for simplifying rational functions. To factor a quadratic polynomial, like \(x^2 + 4x + 3\), look for two numbers that multiply to the constant term and add up to the linear coefficient.

• The constant term in our example is 3.
• The linear coefficient is 4.

The factors of 3 that add to 4 are 3 and 1. This means we can rewrite it as \((x+3)(x+1)\). This simplification step is crucial for finding vertical asymptotes and holes in the graph.
Once the polynomial is factored, you'll often notice common factors between the numerator and denominator, spaces where potential x-interruption lies.

Graphing Rational Functions

Graphing rational functions begins with understanding the simplified form of the function. Once a rational function is simplified by factoring, you graph the remaining function, being mindful of any restrictions.
For example, after canceling a common factor like \(x+3\) in the given function, the simplified form becomes \(f(x) = x + 1\). This is a simple linear function, but with a twist due to restrictions from the original function.

• With rational function graphs, always check for non-obvious restrictions that might cause a hole.
• Traditional graph features such as slope and y-intercept should be noted.

Understanding these restrictions will give a clearer picture of the function and its graph. Thus, graphing rational functions often involves an extra layer of analysis compared to typical functions.

Identifying Holes in Graphs

Holes in the graph appear where a rational function is not defined, typically at x-values that cause both the numerator and denominator to become zero after simplification. For the function \(f(x)=\frac{x^{2}+4x+3}{x+3}\), factoring reveals a common \(x+3\) term in both parts. Canceling results in a simpler expression, yet a point of discontinuity is created at \(x = -3\), since substituting this value into the original function results in a zero denominator.

• To identify holes, look for canceled factors that affect the original domain.
• These are points where the graph doesn't exist or the function has no value.

Ultimately, these holes are important for accurately graphing rational functions since they represent domains of exclusion where the function can't be evaluated.