Complete the following steps for the given functions.

b. Find the vertical asymptotes of $f$ (if any).

$f (x) = \frac{x^{2} - 3}{x + 6}$

Verified step by step guidance

Step 1: Understand the concept of vertical asymptotes. Vertical asymptotes occur where the denominator of a rational function is zero, and the numerator is not zero at the same point.

Step 2: Identify the denominator of the function $ f(x) = \frac{x^2 - 3}{x + 6} $. The denominator is $ x + 6 $.

Step 3: Set the denominator equal to zero to find the potential vertical asymptotes: $ x + 6 = 0 $.

Step 4: Solve the equation $ x + 6 = 0 $ to find the value of $ x $ that makes the denominator zero.

Step 5: Verify that the numerator $ x^2 - 3 $ is not zero at $ x = -6 $ to confirm the vertical asymptote.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator approaches zero while the numerator does not. This results in the function tending towards infinity or negative infinity. To find vertical asymptotes, set the denominator equal to zero and solve for the variable. In the given function, vertical asymptotes indicate values of x where the function is undefined.

Rational Functions

A rational function is a function represented by the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the behavior of rational functions is crucial for analyzing their graphs, including identifying asymptotes, intercepts, and end behavior. The function provided, f(x) = (x^2 - 3)/(x + 6), is a rational function.

Finding Roots

Finding roots of a polynomial involves determining the values of x that make the polynomial equal to zero. For rational functions, roots of the numerator indicate x-intercepts, while roots of the denominator indicate vertical asymptotes. In the context of the given function, identifying the roots of the denominator is essential for locating vertical asymptotes, as these points are where the function is undefined.

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