Domain And Range Of F(x) = -4x² + 16x + 1 In Interval Notation

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Hey guys! Let's dive into the fascinating world of quadratic functions, and today we're going to dissect the function f(x) = -4x² + 16x + 1. We'll figure out its domain and range, which are crucial for understanding how this function behaves. So, grab your thinking caps, and let's get started!

Understanding the Domain

When we talk about the domain of a function, we're essentially asking: what are all the possible x-values that we can plug into this function without causing any mathematical mayhem? For example, we need to avoid things like dividing by zero or taking the square root of a negative number (at least, within the realm of real numbers). Now, our function here is f(x) = -4x² + 16x + 1. This is a polynomial function, specifically a quadratic function, and polynomial functions are incredibly well-behaved. They don't have any of those pesky restrictions that fractions or square roots might introduce. You can plug in any real number for x, and you'll get a real number out. There are no values of x that would make the function explode or become undefined. So, what does that mean for our domain? It means the domain is all real numbers! We can express this in interval notation as (-∞, ∞). This notation simply means that the domain extends infinitely in both the negative and positive directions. In other words, there's no x-value that's off-limits for this quadratic function.

Let's really break down why this is the case for quadratic functions in general. The highest power of x in a quadratic function is 2. This means we're dealing with terms like x² and x, as well as constants. Squaring a number, multiplying it by a constant, or adding constants will never lead to undefined results within the real number system. Think about it: you can square any real number, multiply it by any constant, and add any constants together, and you'll always get a real number as a result. This fundamental property of polynomials is what gives them their unrestricted domain. So, whenever you encounter a polynomial function, you can confidently say that its domain is all real numbers, or (-∞, ∞) in interval notation. This is a great rule of thumb to keep in your mathematical toolkit!

To further solidify this concept, let's consider some examples. If we plug in x = 0 into our function, we get f(0) = -4(0)² + 16(0) + 1 = 1. That's a perfectly valid output. If we plug in x = 1, we get f(1) = -4(1)² + 16(1) + 1 = -4 + 16 + 1 = 13. Again, a valid output. We could try plugging in any number – positive, negative, zero, fractions, decimals – and we'll always get a real number out. This is because the operations involved (squaring, multiplying, adding) are defined for all real numbers. Therefore, the domain of our function f(x) = -4x² + 16x + 1 is indeed all real numbers, represented as (-∞, ∞).

Finding the Range

Now, let's tackle the range of the function. The range is all about the possible y-values, or f(x) values, that our function can produce. Unlike the domain, which is often straightforward for polynomials, the range requires a bit more investigation, especially for quadratic functions. The key here is to recognize that quadratic functions have a parabolic shape. Parabolas are U-shaped curves, and they either open upwards or downwards. This direction, along with the vertex (the turning point of the parabola), will dictate the range.

Our function, f(x) = -4x² + 16x + 1, has a negative coefficient in front of the x² term (-4). This is a crucial piece of information because it tells us that the parabola opens downwards. Think of it like a frown – a negative coefficient makes the parabola “frown.” This means that the parabola has a maximum point, which is the vertex. The y-coordinate of the vertex will be the highest value that the function can reach. All other y-values will be less than or equal to this maximum value. To find the range, we need to determine the y-coordinate of the vertex.

There are a couple of ways to find the vertex. One way is to complete the square. This method involves rewriting the quadratic function in vertex form, which is f(x) = a(x - h)² + k, where (h, k) is the vertex. Another way, which is often quicker, is to use the formula for the x-coordinate of the vertex: x = -b / 2a, where a and b are the coefficients of the quadratic function in the standard form f(x) = ax² + bx + c. In our case, a = -4 and b = 16, so the x-coordinate of the vertex is x = -16 / (2 * -4) = -16 / -8 = 2. Now that we have the x-coordinate of the vertex, we can plug it back into the function to find the y-coordinate: f(2) = -4(2)² + 16(2) + 1 = -4(4) + 32 + 1 = -16 + 32 + 1 = 17. So, the vertex of our parabola is at the point (2, 17).

Since the parabola opens downwards, the vertex represents the maximum point of the function. This means that the maximum y-value that the function can reach is 17. The function will take on all y-values less than or equal to 17. Therefore, the range of the function is (-∞, 17]. The parenthesis on the negative infinity side indicates that the range extends infinitely in the negative direction, while the square bracket on the 17 side indicates that 17 is included in the range. In other words, the function can reach 17, and it can reach any value smaller than 17.

To recap, finding the range of a quadratic function involves understanding the shape of the parabola and locating its vertex. The sign of the coefficient of the x² term tells us whether the parabola opens upwards or downwards. If it opens upwards, the vertex is a minimum point, and the range will be [k, ∞), where k is the y-coordinate of the vertex. If it opens downwards, the vertex is a maximum point, and the range will be (-∞, k]. By finding the vertex and considering the direction of the parabola, we can confidently determine the range of any quadratic function.

Putting It All Together

Okay, guys, let's bring it all together! We've explored the quadratic function f(x) = -4x² + 16x + 1 and successfully determined both its domain and range. We started by understanding that the domain represents all possible x-values that we can plug into the function. Because quadratic functions are polynomials, their domain is always all real numbers, which we express in interval notation as (-∞, ∞). There are no restrictions on the x-values we can use.

Next, we tackled the range, which represents all possible y-values that the function can produce. This required a bit more work because we needed to consider the parabolic shape of the quadratic function. We recognized that the negative coefficient of the x² term meant the parabola opens downwards, indicating a maximum point at the vertex. We then used the formula x = -b / 2a to find the x-coordinate of the vertex, which was 2. Plugging this value back into the function gave us the y-coordinate of the vertex, which was 17. This told us that the maximum y-value the function can reach is 17. Since the parabola opens downwards, the range includes all values less than or equal to 17, which we express in interval notation as (-∞, 17]. So, we have our final answers:

  • Domain: (-∞, ∞)
  • Range: (-∞, 17]

Understanding the domain and range of a function is a fundamental concept in mathematics, and we've successfully applied it to this quadratic function. By breaking down the problem into smaller steps – understanding the definition of domain and range, recognizing the shape of the parabola, and finding the vertex – we were able to solve it effectively. Keep practicing these concepts, and you'll become a pro at analyzing functions in no time!

Remember, the domain tells us what inputs are allowed, while the range tells us what outputs are possible. Together, they give us a complete picture of how a function behaves. And with quadratic functions, understanding the parabolic shape and the vertex is key to unlocking the range. Great job, everyone! You've conquered another mathematical challenge!

Conclusion

In conclusion, we've thoroughly explored the quadratic function f(x) = -4x² + 16x + 1, successfully determining its domain and range. The domain, representing all possible input values, is (-∞, ∞), signifying that any real number can be used as an input. The range, representing all possible output values, is (-∞, 17], indicating that the function's output values extend from negative infinity up to and including 17. By understanding the properties of quadratic functions, particularly their parabolic shape and the significance of the vertex, we can effectively analyze and determine these key characteristics. This process not only enhances our understanding of this specific function but also provides a foundation for analyzing other functions in the future. Keep exploring, keep questioning, and keep learning!