Decoding F(x)=(25-x^2)/(x^2-4x-5) A Comprehensive Analysis

by ADMIN 59 views

Hey guys! Ever stumbled upon a function that looks like it's trying to play hide-and-seek with its asymptotes and values? Well, today we’re diving deep into one such mathematical marvel: f(x) = (25 - x²) / (x² - 4x - 5). We’re going to break down this function piece by piece, uncovering its secrets and understanding its behavior. Buckle up, because it’s going to be a fun ride!

Understanding the Function: A Bird's Eye View

At first glance, this function might seem a bit intimidating, but don't worry! It's just a rational function, which means it's a ratio of two polynomials. The numerator is (25 - x²) and the denominator is (x² - 4x - 5). To truly understand this function, we need to explore its key features, including its domain, intercepts, asymptotes, and any points of discontinuity. So, let's roll up our sleeves and get started.

Factoring is Your Friend

The first step to unraveling this function is to factor both the numerator and the denominator. This will help us identify any common factors, which can lead to holes in the graph, and also make it easier to find the zeros and asymptotes. Factoring the numerator (25 - x²) is like recognizing a classic – it’s a difference of squares! This factors into (5 - x)(5 + x). The denominator (x² - 4x - 5) is a quadratic, which we can factor into (x - 5)(x + 1). Remember, factoring is the key to unlocking many secrets in rational functions!

Simplifying the Function

Now, let’s rewrite our function with the factored forms: f(x) = (5 - x)(5 + x) / (x - 5)(x + 1). Notice anything interesting? Aha! We have a (5 - x) in the numerator and an (x - 5) in the denominator. These are almost the same, but they have opposite signs. We can rewrite (5 - x) as -1(x - 5). This gives us f(x) = -1(x - 5)(5 + x) / (x - 5)(x + 1). Now we can cancel out the (x - 5) terms, but remember, this cancellation comes at a cost – it creates a hole in our graph! After canceling, our simplified function becomes f(x) = -(5 + x) / (x + 1), but we need to keep in mind that x ≠ 5.

Unveiling the Hole

So, what’s this “hole” we’re talking about? When we canceled out the (x - 5) term, we removed a potential vertical asymptote, but we didn’t remove the fact that the original function is undefined at x = 5. This means there’s a hole in the graph at x = 5. To find the y-coordinate of the hole, we plug x = 5 into the simplified function: f(5) = -(5 + 5) / (5 + 1) = -10 / 6 = -5/3. Therefore, there's a hole in the graph at the point (5, -5/3).

Asymptotes: The Invisible Boundaries

Asymptotes are like invisible boundaries that a function approaches but never quite touches. They give us valuable information about the function's behavior as x approaches infinity or specific values. There are three main types of asymptotes: vertical, horizontal, and oblique (or slant). Let's find them for our function.

Vertical Asymptotes: Where the Function Goes Vertical

Vertical asymptotes occur where the denominator of the simplified function equals zero. In our simplified function, f(x) = -(5 + x) / (x + 1), the denominator is (x + 1). Setting this equal to zero gives us x + 1 = 0, which means x = -1. So, we have a vertical asymptote at x = -1. This means that as x approaches -1 from the left or right, the function's value will shoot off towards positive or negative infinity.

Horizontal Asymptotes: The Long-Term Trend

Horizontal asymptotes describe the function's behavior as x approaches positive or negative infinity. To find the horizontal asymptote, we compare the degrees of the numerator and denominator in our simplified function. Both the numerator -(5 + x) and the denominator (x + 1) have a degree of 1. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is -1, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is y = -1/1 = -1. This tells us that as x gets very large (positive or negative), the function's value gets closer and closer to -1.

Oblique Asymptotes: The Slanted Guides

Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In our simplified function, the degrees are equal, so there is no oblique asymptote. We can breathe a sigh of relief – no slanted guides to worry about!

Intercepts: Where the Function Crosses the Axes

Intercepts are the points where the function's graph crosses the x-axis (x-intercepts) and the y-axis (y-intercept). They give us specific points to anchor our understanding of the function's behavior.

X-Intercepts: Crossing the X-Axis

X-intercepts occur where the function's value is zero. To find them, we set the numerator of the simplified function equal to zero: -(5 + x) = 0. This gives us x = -5. So, we have an x-intercept at x = -5, which corresponds to the point (-5, 0).

Y-Intercepts: Meeting the Y-Axis

Y-intercepts occur where x = 0. To find the y-intercept, we plug x = 0 into the simplified function: f(0) = -(5 + 0) / (0 + 1) = -5. Therefore, we have a y-intercept at y = -5, which corresponds to the point (0, -5).

Putting It All Together: Graphing the Function

Now that we've found the asymptotes, intercepts, and hole, we can sketch a graph of the function. We know there’s a vertical asymptote at x = -1, a horizontal asymptote at y = -1, an x-intercept at (-5, 0), a y-intercept at (0, -5), and a hole at (5, -5/3). With this information, we can sketch the curves of the function, making sure they approach the asymptotes and pass through the intercepts, while also remembering to skip over the hole. Graphing is like connecting the dots – we’re bringing all our findings together to visualize the function’s behavior.

Analyzing the Statements: True or False?

Now, let's tackle the original questions. We need to determine which of the following statements are correct:

  1. m ≠ n: This statement is a bit vague without knowing what 'm' and 'n' refer to in this context. However, it might be related to comparing different aspects of the function, such as intercepts or asymptotes. We'll need more context to definitively answer this.
  2. m = n: Similar to the previous statement, we need more context to understand what 'm' and 'n' represent. If they were referring to the number of vertical and horizontal asymptotes, this would be incorrect since we have one of each.
  3. There is only one vertical asymptote: This statement is correct. We found a vertical asymptote at x = -1, and that's the only one.
  4. y = -1 is the horizontal asymptote: This statement is also correct. We determined that the horizontal asymptote is indeed y = -1.

Conclusion: Mastering Rational Functions

Phew! We've taken a deep dive into the function f(x) = (25 - x²) / (x² - 4x - 5), uncovering its factored form, identifying its asymptotes and intercepts, and even spotting a sneaky hole! We analyzed the given statements and determined which ones hold true. By breaking down the function step by step, we've gained a solid understanding of its behavior. Rational functions might seem complex at first, but with a systematic approach and a bit of practice, you can master them like a pro! Keep exploring, keep questioning, and most importantly, keep having fun with math!

Let's clarify the original mathematical questions related to the function f(x)=(25-x2)/(x2-4x-5).

Given the function:

f(x) = (25 - x²) / (x² - 4x - 5)

Which of the following statements are correct? Select all that apply.

  • What do 'm' and 'n' likely represent in the context of comparing different characteristics of the function (e.g., intercepts, asymptotes)? To address the statements "m ≠ n" and "m = n", we need to define the values being compared.
  • Is it true that there is only one vertical asymptote for this function?
  • Is y = -1 the horizontal asymptote of this function?

This revised format provides a clearer understanding of the questions and breaks down the ambiguous statements to facilitate more precise answers.