Asymptotes Of F(x) = (x^2 + X - 6) / (x^3 - 1)

by SLV Team 47 views
Asymptotes of f(x) = (x^2 + x - 6) / (x^3 - 1)

Hey guys! Let's dive into finding the asymptotes of the function f(x) = (x^2 + x - 6) / (x^3 - 1). Asymptotes, those sneaky lines that a function approaches but never quite touches, can tell us a lot about a function's behavior, especially its end behavior and points of discontinuity. We'll break this down step-by-step, making it super clear how to identify both vertical and horizontal asymptotes. So, grab your thinking caps, and let's get started!

Understanding Asymptotes

First, let's make sure we're all on the same page about what asymptotes actually are. Think of them as guide rails for the function's graph.

  • Vertical asymptotes are vertical lines that the graph approaches but never crosses. They typically occur where the function is undefined, like when the denominator of a rational function equals zero.
  • Horizontal asymptotes are horizontal lines that the graph approaches as x goes to positive or negative infinity. They describe the function's behavior at its extreme ends.

Why are asymptotes important, you ask? Well, they give us crucial information about a function's graph without having to plot a million points. They help us understand where the function might have discontinuities (vertical asymptotes) and how it behaves as x gets super large or super small (horizontal asymptotes). Plus, knowing the asymptotes can make sketching the graph so much easier. So, let's find out how to spot them!

Finding Vertical Asymptotes

To find vertical asymptotes, we need to identify where the function is undefined. For rational functions (that's functions that are fractions with polynomials), this usually means finding the values of x that make the denominator equal to zero. However, there's a slight twist: we need to make sure those values don't also make the numerator zero, because that could lead to a hole (a removable discontinuity) instead of a vertical asymptote. So, here's the plan:

  1. Factor both the numerator and the denominator: Factoring helps us see the roots (zeros) of each polynomial clearly. It's like dissecting the function to understand its building blocks.
  2. Set the denominator equal to zero and solve for x: This will give us the potential locations of vertical asymptotes.
  3. Check if the roots of the denominator are also roots of the numerator: If a value of x makes both the numerator and denominator zero, it might be a hole instead of an asymptote. We'll need to simplify the function by canceling out common factors to be sure.
  4. The values of x that make the simplified denominator zero are the vertical asymptotes: These are the vertical lines that our function will approach but never cross.

Let's apply this to our function, f(x) = (x^2 + x - 6) / (x^3 - 1).

Step 1: Factor the Numerator and Denominator

The numerator, x^2 + x - 6, can be factored into (x + 3)(x - 2). This means the numerator equals zero when x = -3 or x = 2.

The denominator, x^3 - 1, is a difference of cubes, which has a special factoring pattern: a^3 - b^3 = (a - b)(a^2 + ab + b^2). In our case, this factors to (x - 1)(x^2 + x + 1). So, the denominator equals zero when x - 1 = 0 or x^2 + x + 1 = 0.

Step 2: Solve for Potential Vertical Asymptotes

Setting the factored denominator equal to zero gives us:

(x - 1)(x^2 + x + 1) = 0

This equation is satisfied if either (x - 1) = 0 or (x^2 + x + 1) = 0.

  • x - 1 = 0 gives us x = 1.

  • x^2 + x + 1 = 0 is a quadratic equation. We can use the quadratic formula to find its roots: x = [-b ± √(b^2 - 4ac)] / (2a). In this case, a = 1, b = 1, and c = 1. Plugging these values in, we get:

    x = [-1 ± √(1^2 - 4 * 1 * 1)] / (2 * 1) x = [-1 ± √(-3)] / 2

    Since the discriminant (the value inside the square root) is negative, this quadratic has no real roots. That means x^2 + x + 1 never equals zero for any real number.

So, the only potential vertical asymptote we've found so far is x = 1.

Step 3: Check for Common Roots

Now, we need to see if x = 1 is also a root of the numerator. Remember, the numerator factored to (x + 3)(x - 2). Plugging in x = 1, we get (1 + 3)(1 - 2) = (4)(-1) = -4, which is not zero. This means x = 1 is not a root of the numerator.

Step 4: Identify Vertical Asymptotes

Since x = 1 makes the denominator zero but not the numerator, it is indeed a vertical asymptote. So, we've found our vertical asymptote: x = 1.

Finding Horizontal Asymptotes

Horizontal asymptotes tell us what happens to the function's value (y) as x becomes very large (positive infinity) or very small (negative infinity). There's a handy rule for finding horizontal asymptotes of rational functions that depends on the degrees of the numerator and denominator polynomials:

  • Case 1: Degree of numerator < Degree of denominator: The horizontal asymptote is y = 0. Think of it like the denominator growing much faster than the numerator, squishing the fraction towards zero.
  • Case 2: Degree of numerator = Degree of denominator: The horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator). The leading coefficients are the numbers in front of the highest power of x.
  • Case 3: Degree of numerator > Degree of denominator: There is no horizontal asymptote. The function will either go to positive or negative infinity as x goes to infinity.

Let's apply this to our function, f(x) = (x^2 + x - 6) / (x^3 - 1).

Determine the Degrees

The degree of the numerator (x^2 + x - 6) is 2 (the highest power of x is x^2). The degree of the denominator (x^3 - 1) is 3 (the highest power of x is x^3).

Apply the Rules

Since the degree of the numerator (2) is less than the degree of the denominator (3), we fall into Case 1. Therefore, the horizontal asymptote is y = 0.

Putting It All Together

Okay, we've done the work! Let's recap our findings:

  • Vertical Asymptote: x = 1
  • Horizontal Asymptote: y = 0

So, the correct answer is B. vertical asymptote: x = 1, horizontal asymptote: y = 0

Conclusion

Finding asymptotes might seem a bit tricky at first, but with a clear understanding of the rules and a step-by-step approach, you can totally master it! Remember to factor, set the denominator to zero, check for common roots, and compare the degrees of the numerator and denominator. Asymptotes are powerful tools for understanding and sketching graphs, so keep practicing, and you'll be spotting them like a pro in no time! Keep up the great work, guys!