Function Analysis: Domain, Properties & Variability
Hey guys! Let's dive into the world of function analysis, breaking down a specific problem step-by-step. We'll be investigating the function f(x) = -2x² / (x+1)², exploring its domain, key properties, and overall behavior. Think of this as our mathematical treasure hunt, where each step unveils a new piece of the puzzle.
1. Determining the Domain of the Function
So, first things first, let's figure out the domain of our function, f(x) = -2x² / (x+1)². In simple terms, the domain is the set of all possible input values (x-values) for which the function produces a valid output. We need to identify any values of x that would make our function go haywire, leading to undefined results.
Looking at our function, we have a rational expression – a fraction. The main concern with fractions is the denominator. We know that division by zero is a big no-no in mathematics because it's undefined. Therefore, we need to find any x values that would make the denominator, (x+1)², equal to zero. Let's set the denominator to zero and solve for x:
(x + 1)² = 0
Taking the square root of both sides, we get:
x + 1 = 0
Subtracting 1 from both sides:
x = -1
Aha! We've found our culprit. If x = -1, the denominator becomes zero, and our function is undefined. This means x = -1 is the only value we need to exclude from our domain. Therefore, the domain of the function f(x) = -2x² / (x+1)² is all real numbers except -1. We can express this in a few ways:
- Set notation: { x ∈ ℝ | x ≠ -1 }
- Interval notation: (-∞, -1) ∪ (-1, ∞)
Think of it like this: imagine a number line stretching infinitely in both directions. Our domain includes every single point on that line except for the point at x = -1. We've created a little gap there to avoid the undefined territory of division by zero. Determining the domain is super crucial because it sets the stage for our entire analysis. It tells us where our function is actually "alive" and kicking, allowing us to explore its behavior within those boundaries. Without a clear understanding of the domain, we might end up drawing incorrect conclusions about the function's properties. So, pat yourselves on the back, guys, you've conquered the first step!
2. Basic Properties of the Function
Now that we've nailed down the domain, let's dig into the basic properties of our function, f(x) = -2x² / (x+1)². This is where we'll uncover some key characteristics that define its behavior. We'll be looking at evenness/oddness, periodicity, zeros (where the function crosses the x-axis), and intersection points (where it crosses the y-axis). Think of these as the function's vital statistics – they tell us a lot about its personality.
Evenness or Oddness
First up, let's explore whether our function is even, odd, or neither. This property relates to the function's symmetry. An even function is symmetrical about the y-axis, meaning if you fold the graph along the y-axis, the two halves would perfectly overlap. Mathematically, this means f(-x) = f(x) for all x in the domain. An odd function, on the other hand, exhibits rotational symmetry about the origin. If you rotate the graph 180 degrees around the origin, it'll look the same. Mathematically, this translates to f(-x) = -f(x) for all x in the domain.
To test our function, f(x) = -2x² / (x+1)², we need to find f(-x). Let's substitute -x for x in our function:
f(-x) = -2(-x)² / (-x+1)² = -2x² / (1-x)²
Now, let's compare f(-x) with f(x). We have:
f(x) = -2x² / (x+1)²
f(-x) = -2x² / (1-x)²
Notice that the numerators are the same, but the denominators are different: (x+1)² versus (1-x)². These aren't the same in general. For example, if we plug in x = 2, we get (2+1)² = 9 and (1-2)² = 1. Since f(-x) is not equal to f(x), our function is not even. Also, f(-x) is not equal to -f(x), so it's not an odd function either. Therefore, our function f(x) = -2x² / (x+1)² is neither even nor odd. This means it doesn't possess either y-axis symmetry or rotational symmetry about the origin.
Periodicity
Next, let's investigate periodicity. A periodic function repeats its values at regular intervals. Think of a wave that goes up and down, repeating the same pattern over and over. Mathematically, a function f(x) is periodic if there exists a non-zero constant P (the period) such that f(x + P) = f(x) for all x in the domain.
Our function, f(x) = -2x² / (x+1)², is not a trigonometric function (like sine or cosine), which are the usual suspects for periodic behavior. Rational functions like ours typically aren't periodic. To confirm this, we can try to find a period P that satisfies the condition f(x + P) = f(x). However, substituting (x + P) into our function and trying to simplify the equation will quickly show that there's no such constant P. Therefore, we can confidently say that f(x) = -2x² / (x+1)² is not periodic. It doesn't have a repeating pattern over any interval.
Zeros of the Function
Now, let's find the zeros of the function. These are the x-values where the function crosses the x-axis, meaning f(x) = 0. To find them, we set our function equal to zero and solve for x:
-2x² / (x+1)² = 0
A fraction is equal to zero if and only if its numerator is zero (and the denominator is non-zero). So, we focus on the numerator:
-2x² = 0
Dividing both sides by -2:
x² = 0
Taking the square root of both sides:
x = 0
We found a zero! Our function has one zero at x = 0. This means the graph of the function touches the x-axis at the point (0, 0). The function's zeros are crucial for understanding where the graph crosses or touches the x-axis, giving us key anchor points for sketching its shape. In this case, knowing that our function has a zero at x = 0 tells us a significant piece of its behavior near the origin.
Intersection Points
Finally, let's determine the intersection points, specifically the y-intercept. This is the point where the function's graph intersects the y-axis. It occurs when x = 0. Luckily, we already found this when we were looking for the zeros! We know that f(0) = -2(0)² / (0+1)² = 0. So, the y-intercept is at the point (0, 0).
We've now successfully explored the basic properties of our function. We know it's neither even nor odd, it's not periodic, and it has a zero and y-intercept at x = 0. These properties provide us with a solid foundation for further analysis, giving us clues about the function's symmetry, repeating patterns, and key points on its graph. You guys are doing awesome! We're piecing together the puzzle, one property at a time.
In conclusion, by systematically examining these properties, we gain a deeper understanding of the function's behavior and can visualize its graph more accurately. This comprehensive approach lays a strong groundwork for more advanced analysis, such as finding local maxima and minima, intervals of increase and decrease, and concavity.