Analyzing F(x) = 2x^3 - 7x^2 + 6: Domain, Range, And Behavior

Hey guys! Let's dive into analyzing the cubic function f(x) = 2x^3 - 7x^2 + 6. We'll break down its key features, including its domain, range, relative extrema, end behavior, and intervals of increase and decrease. Understanding these aspects will give us a solid grasp of how this function behaves.

Domain

Let's start with the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions, like our cubic function f(x), there are no restrictions on the input values. You can plug in any real number for x, and you'll get a real number output. There are no denominators that could be zero, no square roots of negative numbers, or any other funky stuff that would limit our inputs. So, what does this mean for our function? Well...

The domain of f(x) = 2x^3 - 7x^2 + 6 is all real numbers. We can write this in a couple of ways: using interval notation, we say the domain is (-∞, ∞), and using set notation, we say it's {x | x ∈ ℝ}, where ℝ represents the set of all real numbers. This is awesome because it means we don't have to worry about any sneaky restrictions on what we can plug into our function. We can explore the entire number line, both positive and negative, and everything in between. This unrestricted domain is a hallmark of polynomial functions, making them super versatile and predictable in many ways. So, whenever you see a polynomial, you can confidently say, "Hey, its domain is all real numbers!" It's one less thing to worry about when you're diving into the deeper analysis of the function's behavior. Now that we've nailed down the domain, let's move on to another crucial aspect: the range.

Range

Now, let's figure out the range. The range of a function is the set of all possible output values (y-values) that the function can produce. Unlike the domain, the range can be a bit trickier to determine, especially for polynomial functions. But don't worry, we'll tackle it together! For cubic functions, like ours, the range is also all real numbers. Think about it this way: as x gets incredibly large in the positive direction (approaching positive infinity), the 2x^3 term will dominate the function, causing f(x) to also head towards positive infinity. Similarly, as x gets incredibly large in the negative direction (approaching negative infinity), the 2x^3 term will again dominate, but this time, f(x) will head towards negative infinity.

What does this tell us? It tells us that the function covers the entire vertical number line. There's no upper or lower bound on the possible output values. No matter how large or small a y-value you pick, you can find an x-value that will produce it. This is a characteristic feature of odd-degree polynomials (like cubics) – they stretch from negative infinity to positive infinity in their range. So, just like the domain, the range of f(x) = 2x^3 - 7x^2 + 6 is all real numbers, expressed as (-∞, ∞) in interval notation or {y | y ∈ ℝ} in set notation. This is great news! We've got another key piece of the puzzle solved. We know the function can take any input and produce any output across the entire number line. With the domain and range settled, let's move on to something a little more exciting: finding the relative maxima and minima of the function. These are the peaks and valleys of our cubic rollercoaster, and they'll give us some serious insights into its shape and behavior.

Relative Maxima and Minima

Next up, let's find those relative maxima and minima – the local peaks and valleys of our function. To do this, we'll need to use a little calculus. First, we find the derivative of f(x). Remember, the derivative tells us the slope of the tangent line at any point on the function, and the relative extrema occur where the slope is zero (or undefined, but that's not a concern for polynomials). So, let's roll up our sleeves and differentiate:

f(x) = 2x^3 - 7x^2 + 6

Using the power rule, we get:

f'(x) = 6x^2 - 14x

Now, we set the derivative equal to zero and solve for x:

6x^2 - 14x = 0

We can factor out a 2x:

2x(3x - 7) = 0

This gives us two critical points:

x = 0 and x = 7/3

These are the x-values where our function might have a relative maximum or minimum. To determine whether they are maxima or minima (or neither), we can use the second derivative test. Let's find the second derivative:

f'(x) = 6x^2 - 14x

f''(x) = 12x - 14

Now, we evaluate the second derivative at our critical points:

For x = 0:

f''(0) = 12(0) - 14 = -14

Since f''(0) is negative, we have a relative maximum at x = 0. To find the y-value, we plug x = 0 back into the original function:

f(0) = 2(0)^3 - 7(0)^2 + 6 = 6

So, we have a relative maximum at the point (0, 6).

For x = 7/3:

f''(7/3) = 12(7/3) - 14 = 28 - 14 = 14

Since f''(7/3) is positive, we have a relative minimum at x = 7/3. To find the y-value, we plug x = 7/3 back into the original function:

f(7/3) = 2(7/3)^3 - 7(7/3)^2 + 6 = -616/27 + 6 = -22.81 + 6 = -16.81

So, we have a relative minimum at the point (7/3, -16.81). Fantastic! We've located the peaks and valleys of our function. We know it has a local high point at (0, 6) and a local low point at approximately (2.33, -16.81). These points are crucial landmarks in understanding the function's overall shape. Now that we've conquered the relative extrema, let's shift our focus to how the function behaves as x goes to extremes – its end behavior.

End Behavior

Let's talk about end behavior. This describes what happens to the function f(x) as x approaches positive and negative infinity. For polynomial functions, the end behavior is primarily determined by the leading term, which in our case is 2x^3. As x gets super large (approaches positive infinity), the 2x^3 term will dominate, and since it has a positive coefficient, f(x) will also head towards positive infinity.

In mathematical notation, we write:

As x → ∞, f(x) → ∞

On the flip side, as x gets extremely negative (approaches negative infinity), the 2x^3 term will still dominate, but because we're cubing a negative number, the result will be negative. So, f(x) will head towards negative infinity.

As x → -∞, f(x) → -∞

This end behavior is typical for cubic functions with a positive leading coefficient. They rise to the right and fall to the left, like a tilted rollercoaster track. Understanding the end behavior is crucial for visualizing the overall shape of the function. It tells us the long-term trends, what happens way out on the fringes of the graph. We know our function climbs skyward as we move to the right and plummets downward as we move to the left. Armed with this knowledge, let's move on to the final pieces of our puzzle: the intervals where the function is increasing and decreasing. This will give us a complete picture of how the function changes direction and shape across its entire domain.

Intervals of Increase and Decrease

Alright, let's figure out where our function is increasing and decreasing. This will help us understand the function's overall shape and how it changes direction. We'll use the first derivative, f'(x) = 6x^2 - 14x, which we found earlier, to determine these intervals. Remember, the first derivative tells us about the slope of the function. Where f'(x) is positive, the function is increasing, and where f'(x) is negative, the function is decreasing. We already found the critical points where f'(x) = 0: x = 0 and x = 7/3. These points divide the x-axis into three intervals that we need to test:

1. (-∞, 0)
2. (0, 7/3)
3. (7/3, ∞)

Let's pick a test value within each interval and plug it into f'(x) to see if it's positive or negative.

Interval 1: (-∞, 0)

Let's pick x = -1:

f'(-1) = 6(-1)^2 - 14(-1) = 6 + 14 = 20

Since f'(-1) is positive, the function is increasing on the interval (-∞, 0).

Interval 2: (0, 7/3)

Let's pick x = 1:

f'(1) = 6(1)^2 - 14(1) = 6 - 14 = -8

Since f'(1) is negative, the function is decreasing on the interval (0, 7/3).

Interval 3: (7/3, ∞)

Let's pick x = 3:

f'(3) = 6(3)^2 - 14(3) = 54 - 42 = 12

Since f'(3) is positive, the function is increasing on the interval (7/3, ∞).

So, to recap:

• Increasing Intervals: (-∞, 0) and (7/3, ∞)
• Decreasing Interval: (0, 7/3)

Awesome! We've now mapped out where our function is climbing uphill and sliding downhill. It increases from negative infinity up to x = 0, then decreases until x = 7/3, and finally increases again as it heads towards positive infinity. This perfectly aligns with our findings about the relative maxima and minima – a maximum at x = 0 and a minimum at x = 7/3. This consistency in our analysis gives us confidence that we've accurately captured the function's behavior.

Conclusion

Alright, guys, we've done it! We've thoroughly analyzed the function f(x) = 2x^3 - 7x^2 + 6, and we now have a comprehensive understanding of its behavior. We found the domain and range (both all real numbers), the relative maximum at (0, 6), the relative minimum at (7/3, -16.81), the end behavior (as x approaches infinity, f(x) approaches infinity, and as x approaches negative infinity, f(x) approaches negative infinity), and the intervals of increase and decrease (increasing on (-∞, 0) and (7/3, ∞), and decreasing on (0, 7/3)). By breaking down the function into these key components, we've gained a deep understanding of its shape and characteristics. This kind of detailed analysis is super valuable in calculus and beyond, helping us to model and understand real-world phenomena. Keep up the great work, and remember, practice makes perfect! The more functions you analyze, the better you'll become at spotting patterns and understanding their behavior. Now go forth and conquer those cubic functions!