Domain And Range Of H(x) = -(x-2)^2 + 3 A Comprehensive Guide

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Understanding the domain and range of a function is a fundamental concept in mathematics. The domain refers to the set of all possible input values (x-values) for which the function is defined, while the range refers to the set of all possible output values (y-values or h(x) values) that the function can produce. In this article, we will delve into determining the domain and range of the quadratic function h(x) = -(x-2)^2 + 3. This process involves analyzing the function's structure, identifying any restrictions on the input values, and understanding how the function transforms the input values to produce the output values.

Domain of h(x) = -(x-2)^2 + 3

When determining the domain of a function, the primary question to address is: Are there any input values (x-values) that would make the function undefined or impossible to compute? Common scenarios that lead to restrictions on the domain include division by zero, taking the square root of a negative number, or taking the logarithm of a non-positive number. However, in the case of the function h(x) = -(x-2)^2 + 3, we encounter none of these restrictions. This function is a quadratic function, which is a type of polynomial function. Polynomial functions are defined for all real numbers, meaning that any real number can be plugged in for x and the function will produce a valid output.

To further clarify, let's break down the function:

  • (x-2): This part involves subtracting 2 from x. Subtraction is defined for all real numbers.
  • (x-2)^2: This part involves squaring the result of (x-2). Squaring is also defined for all real numbers. Any real number squared will result in a non-negative real number.
  • -(x-2)^2: This part involves taking the negative of the squared term. Multiplying by -1 is defined for all real numbers.
  • -(x-2)^2 + 3: This part involves adding 3 to the negative squared term. Addition is defined for all real numbers.

Since each operation within the function is defined for all real numbers, the entire function h(x) = -(x-2)^2 + 3 is also defined for all real numbers. Therefore, there are no restrictions on the input values. We can express the domain of h(x) using interval notation as (-∞, ∞), which signifies that the domain includes all real numbers from negative infinity to positive infinity. Alternatively, we can express the domain using set notation as {x | x ∈ ℝ}, which reads as "the set of all x such that x is an element of the set of real numbers."

In summary, the domain of h(x) = -(x-2)^2 + 3 is the set of all real numbers. This is a characteristic property of polynomial functions, which are generally well-behaved and defined across the entire real number line. Recognizing this fundamental aspect of polynomial functions simplifies the process of determining their domains, allowing us to focus on other aspects of their behavior, such as their range and graphical representation.

Range of h(x) = -(x-2)^2 + 3

Determining the range of a function, which is the set of all possible output values (y-values or h(x) values), often requires a different approach than determining the domain. For the function h(x) = -(x-2)^2 + 3, we need to consider how the transformations applied to x affect the possible output values. The function is a quadratic function in vertex form, which provides valuable information about its shape and maximum or minimum value.

The general form of a quadratic function in vertex form is:

f(x) = a(x-h)^2 + k

Where:

  • (h, k) represents the vertex of the parabola, which is the point where the function reaches its maximum or minimum value.
  • 'a' determines the direction the parabola opens (upwards if a > 0, downwards if a < 0) and the vertical stretch or compression.

Comparing h(x) = -(x-2)^2 + 3 to the vertex form, we can identify the following:

  • a = -1
  • h = 2
  • k = 3

This tells us that the vertex of the parabola is at the point (2, 3). Since 'a' is negative (-1), the parabola opens downwards. This means that the vertex represents the maximum point of the function. The maximum value of the function is the y-coordinate of the vertex, which is 3.

Now, let's analyze how the transformations affect the range:

  • (x-2)^2: As mentioned earlier, squaring any real number results in a non-negative value. Therefore, (x-2)^2 will always be greater than or equal to 0.
  • -(x-2)^2: Multiplying a non-negative value by -1 results in a non-positive value (a value less than or equal to 0). Therefore, -(x-2)^2 will always be less than or equal to 0.
  • -(x-2)^2 + 3: Adding 3 to a non-positive value will result in a value less than or equal to 3. Therefore, h(x) = -(x-2)^2 + 3 will always be less than or equal to 3.

This confirms that the maximum value of h(x) is 3. Since the parabola opens downwards, the function will take on all y-values less than or equal to 3. There is no lower bound on the y-values, as the parabola extends downwards indefinitely.

Therefore, the range of h(x) = -(x-2)^2 + 3 is all real numbers less than or equal to 3. We can express this using interval notation as (-∞, 3], where the square bracket indicates that 3 is included in the range. Alternatively, we can use set notation as {y | y ≤ 3}, which reads as "the set of all y such that y is less than or equal to 3."

In conclusion, by analyzing the vertex form of the quadratic function and considering the effect of each transformation, we have determined that the range of h(x) = -(x-2)^2 + 3 is (-∞, 3]. This means that the function's output values will never exceed 3, and it will cover all values below 3.

Graphical Representation and Interpretation

Visualizing the graph of the function h(x) = -(x-2)^2 + 3 can provide a valuable understanding of its domain and range. The graph of a quadratic function is a parabola, and in this case, it's a parabola that opens downwards due to the negative coefficient of the squared term. The vertex of the parabola, as we determined earlier, is at the point (2, 3).

When we plot the graph, we observe the following:

  • The parabola extends infinitely to the left and right along the x-axis. This visually confirms that the domain of the function is all real numbers, (-∞, ∞), as there are no restrictions on the x-values.
  • The highest point on the parabola is the vertex (2, 3). The parabola opens downwards, so all other points on the graph have y-values less than 3. This visually confirms that the range of the function is all real numbers less than or equal to 3, (-∞, 3].

Furthermore, the graph helps us understand the relationship between the domain and range. For every x-value we choose on the x-axis, there is a corresponding point on the parabola. The y-coordinate of that point represents the function's output value for that particular x-value. By tracing the parabola, we can see how the y-values vary as we move along the x-axis. The graph clearly demonstrates that the y-values are bounded above by 3, which is the y-coordinate of the vertex, and extend downwards without limit.

The graphical representation also reinforces the concept of the vertex as the maximum point of the function. The vertex is the turning point of the parabola, and since the parabola opens downwards, it represents the highest point the function reaches. This visual confirmation of the maximum value being 3 aligns with our analytical determination of the range.

In addition to understanding the domain and range, the graph can also provide insights into other properties of the function, such as its symmetry. Parabolas are symmetrical about a vertical line that passes through the vertex. In this case, the axis of symmetry is the line x = 2. This means that the function's values are the same for x-values that are equidistant from x = 2. For example, h(1) and h(3) will have the same value, as both 1 and 3 are 1 unit away from 2.

In summary, the graphical representation of h(x) = -(x-2)^2 + 3 provides a visual confirmation of the domain and range we determined analytically. It also enhances our understanding of the function's behavior and properties, such as its maximum value and symmetry. The graph serves as a powerful tool for interpreting and communicating mathematical concepts.

Practical Applications of Domain and Range

Understanding the domain and range of a function is not just a theoretical exercise; it has significant practical applications in various fields. In real-world scenarios, mathematical functions are often used to model relationships between different quantities. The domain and range of these functions represent the realistic constraints and possible outcomes within the context of the problem.

For instance, consider a scenario where the function h(x) = -(x-2)^2 + 3 models the height of a projectile (in meters) above the ground at time x (in seconds) after it is launched. In this context, the domain and range have specific physical interpretations:

  • Domain: The domain represents the time interval during which the projectile is in the air. Since time cannot be negative, the domain would be restricted to x ≥ 0. Furthermore, the projectile will eventually hit the ground, so there will be an upper limit on the time it is in the air. To find this upper limit, we would need to determine when h(x) = 0 (the height is zero). In this practical context, the domain represents the realistic time frame for the projectile's motion.
  • Range: The range represents the possible heights the projectile reaches above the ground. As we determined earlier, the maximum height is 3 meters (the vertex of the parabola). Since the height cannot be negative, the range would be restricted to 0 ≤ h(x) ≤ 3. In this practical context, the range provides information about the vertical extent of the projectile's trajectory.

Another example can be found in economics. Suppose a function models the profit of a company based on the number of units produced. The domain would represent the feasible number of units the company can produce, which might be limited by factors such as production capacity or available resources. The range would represent the possible profit values, which could be limited by factors such as market demand or production costs.

In general, when applying mathematical functions to real-world problems, it's crucial to consider the practical implications of the domain and range. The domain ensures that the input values are meaningful and realistic within the context of the problem, while the range provides information about the possible output values and their limitations.

Furthermore, understanding the domain and range can help in identifying potential issues with the model. For example, if the model predicts an output value outside the realistic range, it might indicate that the model needs to be refined or that certain assumptions are not valid. Similarly, if the domain is too restrictive, it might suggest that the model does not capture the full range of possibilities.

In conclusion, the concepts of domain and range are essential for applying mathematical functions to real-world problems. They provide a framework for understanding the limitations and possibilities within the context of the problem, ensuring that the model is used appropriately and that the results are interpreted meaningfully. By carefully considering the domain and range, we can gain valuable insights into the relationships between different quantities and make informed decisions based on mathematical models.

Conclusion

In this comprehensive exploration, we have thoroughly investigated the process of determining the domain and range of the quadratic function h(x) = -(x-2)^2 + 3. We began by defining the core concepts of domain and range, emphasizing their importance in understanding the behavior and limitations of functions. We then meticulously analyzed the function h(x), considering its structure and the transformations applied to the input variable x.

For the domain, we established that the function is defined for all real numbers. This conclusion was reached by recognizing that the function is a polynomial, and polynomials are generally defined across the entire real number line. We also examined the individual operations within the function (subtraction, squaring, negation, and addition) to confirm that none of them impose any restrictions on the input values. Therefore, the domain of h(x) is (-∞, ∞), encompassing all real numbers.

For the range, we employed a different approach, focusing on the vertex form of the quadratic function. By identifying the vertex as (2, 3) and recognizing that the parabola opens downwards (due to the negative coefficient of the squared term), we deduced that the maximum value of the function is 3. We then analyzed how the transformations affect the output values, confirming that h(x) will always be less than or equal to 3. Consequently, the range of h(x) is (-∞, 3], including all real numbers less than or equal to 3.

We further enhanced our understanding by examining the graphical representation of the function. The graph visually confirmed our analytical findings, demonstrating that the parabola extends infinitely along the x-axis (domain) and reaches a maximum height of 3 (range). The graph also provided insights into the function's symmetry and the role of the vertex as the maximum point.

Finally, we explored the practical applications of domain and range in real-world scenarios. We discussed how these concepts are crucial for interpreting mathematical models and ensuring that the input and output values are meaningful within the context of the problem. Examples from projectile motion and economics illustrated how the domain and range can represent realistic constraints and possible outcomes.

In summary, determining the domain and range of a function is a fundamental skill in mathematics with wide-ranging applications. By combining analytical techniques, graphical representations, and practical considerations, we can gain a deep understanding of a function's behavior and its relevance to real-world problems. The specific case of h(x) = -(x-2)^2 + 3 served as a valuable example for illustrating these concepts and techniques. This knowledge empowers us to effectively use mathematical models and make informed decisions based on their predictions.