Domain And Range Of F(x)=-x²-2x+15 Explained

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Understanding the domain and range of a function is a fundamental concept in mathematics, especially when dealing with quadratic functions. In this article, we will explore the quadratic function $f(x) = -x^2 - 2x + 15$, meticulously examining its domain and range. We'll dissect the equation, analyze its graphical representation, and employ various mathematical techniques to arrive at the correct answer. This comprehensive guide aims to provide a clear and thorough understanding of how to determine the domain and range of such functions.

Understanding the Domain of f(x) = -x² - 2x + 15

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's all the values you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number. For polynomial functions, like our quadratic function $f(x) = -x^2 - 2x + 15$, the domain is generally all real numbers. This is because you can substitute any real number into the equation, and it will produce a valid output. There are no restrictions imposed by the nature of the quadratic expression itself.

To further clarify, let's consider why this is the case. The function involves squaring a number (-x²), multiplying by a constant (-2x), and adding constants (+15). None of these operations impose any limitations on the values of x. You can square any real number, multiply it by any constant, and add constants without encountering any mathematical inconsistencies. Therefore, the domain of $f(x) = -x^2 - 2x + 15$ extends across the entire number line, encompassing all real numbers. Mathematically, this is represented as $(-\infty, \infty)$, indicating that x can take any value from negative infinity to positive infinity.

The concept of the domain is crucial because it sets the stage for understanding the behavior of the function. It tells us over what interval the function is valid and where we can expect to find meaningful outputs. In the context of real-world applications, the domain might represent physical constraints or limitations. For instance, if our function modeled the height of a projectile, the domain might be restricted to non-negative time values. However, in the purely mathematical context of our quadratic function, the domain remains unrestricted, allowing us to explore the full scope of its behavior.

In conclusion, the domain of the quadratic function $f(x) = -x^2 - 2x + 15$ is all real numbers. This understanding forms the foundation for our next step: determining the range of the function.

Determining the Range of f(x) = -x² - 2x + 15

While the domain tells us the possible input values, the range of a function defines the set of all possible output values (y-values) that the function can produce. For the quadratic function $f(x) = -x^2 - 2x + 15$, determining the range requires a bit more analysis than determining the domain. Since the function is a parabola, its range will be limited either above or below, depending on whether the parabola opens upwards or downwards. In our case, the coefficient of the $x^2$ term is negative (-1), indicating that the parabola opens downwards. This means the function has a maximum value, and the range will consist of all y-values less than or equal to this maximum.

To find the maximum value, we need to determine the vertex of the parabola. The vertex represents the highest point on the graph when the parabola opens downwards. The x-coordinate of the vertex can be found using the formula $x = -b / (2a)$, where a and b are the coefficients of the $x^2$ and x terms, respectively. In our function, $a = -1$ and $b = -2$, so the x-coordinate of the vertex is $x = -(-2) / (2 * -1) = -1$.

Now, we can find the y-coordinate of the vertex by substituting this x-value back into the function: $f(-1) = -(-1)^2 - 2(-1) + 15 = -1 + 2 + 15 = 16$. This tells us that the vertex of the parabola is at the point (-1, 16). Since the parabola opens downwards, the maximum value of the function is 16. Therefore, the range of the function consists of all y-values less than or equal to 16.

Mathematically, we represent the range as ${y | y \leq 16}$. This notation signifies the set of all y-values such that y is less than or equal to 16. This range is a direct consequence of the parabolic nature of the quadratic function and the negative coefficient of the $x^2$ term. The vertex, as the highest point on the graph, dictates the upper bound of the range. Understanding the range provides valuable insights into the function's behavior, allowing us to predict the possible outputs and interpret the function in various contexts.

In summary, the range of the quadratic function $f(x) = -x^2 - 2x + 15$ is ${y | y \leq 16}$. This, combined with our earlier determination of the domain, gives us a complete picture of the function's input-output relationship.

Visualizing the Domain and Range Graphically

A graphical representation of the function $f(x) = -x^2 - 2x + 15$ offers a visual confirmation of our calculated domain and range. When we plot this function on a coordinate plane, we observe a parabola that opens downwards. This downward orientation is a direct result of the negative coefficient of the $x^2$ term, as we discussed earlier.

The graph extends infinitely to the left and right along the x-axis, visually demonstrating that the domain is all real numbers. There are no breaks, gaps, or restrictions on the x-values for which the function is defined. This aligns perfectly with our algebraic determination of the domain as $(-\infty, \infty)$. No matter what x-value you choose, you can find a corresponding point on the graph, further solidifying the concept of an unrestricted domain for this quadratic function.

The range, on the other hand, is visually represented by the vertical extent of the parabola. The highest point on the graph, the vertex, is located at the point (-1, 16). This corresponds to the maximum value of the function, which we calculated to be 16. The graph extends downwards from this point, covering all y-values less than or equal to 16. This graphical depiction confirms our calculated range of ${y | y \leq 16}$. The parabola never goes above the y-value of 16, reinforcing the idea that the range is bounded above by this maximum value.

Visualizing the graph provides an intuitive understanding of the domain and range. It bridges the gap between the algebraic representation of the function and its geometric interpretation. The graph serves as a powerful tool for verifying our calculations and for gaining a deeper appreciation of the function's behavior. By observing the graph, we can readily identify the domain as the projection of the curve onto the x-axis and the range as the projection of the curve onto the y-axis. This visual approach is particularly helpful for understanding the concepts of domain and range in a more holistic manner.

In essence, the graphical representation serves as a visual proof of our analytical findings. It reinforces the notion that the domain encompasses all real numbers, while the range is limited to y-values less than or equal to 16. This combination of algebraic and graphical analysis provides a comprehensive understanding of the function $f(x) = -x^2 - 2x + 15$.

Identifying the Correct Answer

Having thoroughly analyzed the function $f(x) = -x^2 - 2x + 15$, we can now confidently identify the correct answer regarding its domain and range. We established that the domain is all real numbers, as there are no restrictions on the x-values that can be input into the function. We also determined that the range is ${y | y \leq 16}$, meaning the function's output values are limited to 16 and below due to the parabola opening downwards and having a maximum value at its vertex.

Based on these findings, we can evaluate the given answer choices. Let's consider the options:

  • A. The domain is all real numbers. The range is ${y | y < 16}$.
  • B. The domain is all real numbers. The range is ${y | y \leq 16}$.
  • C. (This option is intentionally omitted as we have enough information to determine the correct answer.)

By comparing our calculated domain and range with the options, we can see that option B accurately reflects our findings. Option A is close, but it incorrectly states that the range includes y-values strictly less than 16, excluding the value 16 itself. However, we know that the function reaches a maximum value of 16 at its vertex, so 16 must be included in the range.

Therefore, the correct answer is:

  • B. The domain is all real numbers. The range is ${y | y \leq 16}$.

This conclusion is supported by our algebraic calculations, graphical representation, and a thorough understanding of the properties of quadratic functions. By systematically analyzing the function and its characteristics, we were able to confidently arrive at the correct answer.

Conclusion

In conclusion, determining the domain and range of a function is a crucial skill in mathematics. For the quadratic function $f(x) = -x^2 - 2x + 15$, we have demonstrated a comprehensive approach to finding these values. We established that the domain is all real numbers, signifying that any real number can be used as an input for the function. The range, however, is restricted to y-values less than or equal to 16, represented as ${y | y \leq 16}$. This limitation arises from the parabolic nature of the function and its downward-opening orientation.

Our analysis involved both algebraic and graphical methods. We used the formula $x = -b / (2a)$ to find the x-coordinate of the vertex and then substituted this value back into the function to find the maximum y-value, which determined the upper bound of the range. We also visualized the function's graph, which provided a clear representation of the domain and range as the projections of the curve onto the x and y axes, respectively.

The combination of algebraic calculations and graphical analysis provides a robust and intuitive understanding of the function's behavior. By mastering these techniques, you can confidently determine the domain and range of various functions, enhancing your problem-solving abilities in mathematics and related fields. Understanding domain and range is not just a mathematical exercise; it's a fundamental concept that underpins many applications of functions in real-world scenarios.

This comprehensive exploration of the function $f(x) = -x^2 - 2x + 15$ serves as a valuable example of how to approach similar problems. By carefully considering the function's properties, applying appropriate mathematical tools, and verifying your results graphically, you can gain a deep understanding of its domain and range.