Problem 33 - 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. $$ h(x)=\frac{x+7}{x^{2}+4 x-21} $$

Step-by-Step Solution

Answer

The function \(h(x) = \frac{x+7}{x^{2}+4x-21}\) has one vertical asymptote at \(x = 3\) and a hole at \(x = -7\).

Step 1: Simplify Denominator

Begin by simplifying the denominator \(x^{2}+4x-21\) into its factored form. For this, find two numbers that add up to 4 and multiply to -21. The numbers that satisfy these conditions are 7 and -3. Therefore, the factored form of the denominator is \((x-3)(x+7)\). The function \(h(x)\) then becomes \(h(x)=\frac{x+7}{(x-3)(x+7)}\).

Step 2: Identify Vertical Asymptotes

Next, identify the vertical asymptotes. Vertical asymptotes occur when the denominator equals zero but not the numerator. Set the factored denominator equal to zero and solve for x to find the vertical asymptotes: \((x-3)(x+7) = 0\). This yields two solutions, \(x = 3\) and \(x = -7\). However, \(x = -7\) makes both the denominator and the numerator equal to zero, and therefore creates a hole, not an asymptote.

Step 3: Identify Hole

The hole occurs where both the numerator and the denominator of the function equal zero. This happens at \(x = -7\), as both the factor \(x+7\) of the denominator and the \(x+7\) of the numerator are equal to zero at that point. This makes the function undefined at \(x = -7\) and creates a hole there.

Key Concepts

Holes in Rational Functions

Rational functions can sometimes have holes, spots where the function's graph is undefined due to both the numerator and the denominator equaling zero at the same point. This is exactly what happens in some functions when shared factors between the numerator and denominator are set to zero.
To identify holes in rational functions like \( h(x) = \frac{x+7}{(x-3)(x+7)} \), check where both the numerator and denominator have the same zero. In the simplified form, we see that \( x+7 \) is a common factor.
Thus, setting \( x+7 = 0 \) gives \( x = -7 \), which highlights the hole. This means at \( x = -7 \), \( h(x) \) is undefined despite not showing as a vertical asymptote.

• A function will have a hole if a factor in the numerator and denominator cancels out.
• The x-value that makes the common factor zero is where the hole occurs.

Factoring Quadratic Expressions

Factoring quadratic expressions is the key tool used to simplify complicated rational functions and uncover their properties like holes and vertical asymptotes. Understanding how to factor such expressions will lead to better insights into the nature of rational functions.
The standard method involves finding two numbers that sum to the middle coefficient and multiply to the constant term. For the expression \( x^2 + 4x - 21 \), our task is to find numbers that add up to 4 and multiply to -21. These numbers are 7 and -3.
This allows us to rewrite the expression as \((x+7)(x-3)\).

• Remember to check the product ( -21) and the sum (4) of the factors you select.
• Simplifying the expression helps in revealing vertical asymptotes and holes clearly.

Rational Functions

Understanding rational functions is crucial in calculus and advanced algebra as they involve the division of two polynomials. The behavior of these functions is greatly influenced by their denominators and numerators.
A rational function like \( h(x) = \frac{x+7}{x^2+4x-21} \) can be puzzlesome. Still, by factoring the denominator and evaluating how it interacts with the numerator, we can identify important features like vertical asymptotes and holes.
By breaking down \( x^2+4x-21 \) into \((x-3)(x+7)\), it becomes clear that when \( x = -7 \), the function has a hole, while when \( x = 3 \), we have a vertical asymptote.

• Vertical asymptotes indicate where the function tends to infinity.
• Holes occur where both numerator and denominator equal zero.
• Remember to factor the denominator to analyze these behaviors accurately.