Domain And Range Which Function Fits The Criteria

This article delves into the concepts of domain and range in the context of functions, providing a detailed explanation to help you confidently identify the correct function from a given set of options. We will dissect the question: "Which function has a domain of $(-\infty, \infty)$ and a range of $(-\infty, 4]$ ?" and systematically analyze each option to arrive at the accurate answer. Understanding domain and range is crucial not only for solving mathematical problems but also for grasping the behavior and characteristics of various functions.

Decoding Domain and Range

Before we jump into the specific problem, let's solidify our understanding of domain and range. The domain of a function represents the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's the collection of all numbers 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. The range, on the other hand, is the set of all possible output values (y-values) that the function can produce. It represents the span of values the function can attain as you vary the input across its domain. Grasping this distinction is fundamental to solving problems involving functions.

In the given question, we are looking for a function that accepts any real number as input (domain of $(-\infty, \infty)$) and produces output values that are less than or equal to 4 (range of $(-\infty, 4]$). This means the function's graph will extend infinitely to the left and right along the x-axis, and its highest y-value will be 4. The parenthesis in $(-\infty, \infty)$ indicates that negative and positive infinity are not included in the domain, as they are concepts rather than specific numbers. The square bracket in $(-\infty, 4]$ signifies that 4 is included in the range, meaning the function can actually output the value 4.

Analyzing the Options

Now, let's examine each of the provided options in light of our understanding of domain and range:

A. $f(x) = x + 4$

This is a linear function. Linear functions are characterized by a constant rate of change and a straight-line graph. The domain of a linear function is always $(-\infty, \infty)$ because you can input any real number into the equation without encountering any mathematical restrictions. However, the range of this particular linear function is also $(-\infty, \infty)$. As x increases or decreases without bound, so does $f(x)$. There is no upper or lower limit to the output values. Therefore, option A does not satisfy the required range of $(-\infty, 4]$. Linear functions provide a direct relationship between input and output, making their behavior predictable and easy to analyze.

B. $f(x) = -x^2 + 4$

This is a quadratic function. Quadratic functions are defined by a squared term ($x^2$) and their graphs are parabolas. The negative sign in front of the $x^2$ term indicates that the parabola opens downwards. This is a crucial observation because it suggests that the function has a maximum value. The domain of this quadratic function is $(-\infty, \infty)$ because you can square any real number. To determine the range, we need to identify the vertex of the parabola. The vertex represents the maximum or minimum point of the function. In this case, the vertex is at (0, 4). Since the parabola opens downwards, the maximum value of the function is 4. The range is therefore $(-\infty, 4]$. This function perfectly matches the specified domain and range criteria. Understanding quadratic functions is essential in various fields, including physics and engineering, where they model projectile motion and other phenomena.

C. $f(x) = 2^x + 4$

This is an exponential function. Exponential functions have a constant base raised to a variable exponent. The domain of this exponential function is $(-\infty, \infty)$ because you can raise 2 to any real power. However, the range is $(4, \infty)$. The term $2^x$ is always positive, approaching 0 as x goes to negative infinity, and increasing without bound as x goes to positive infinity. Therefore, $2^x + 4$ will always be greater than 4. It will never reach 4, and it will never be less than 4. Thus, option C does not satisfy the required range. Exponential functions are ubiquitous in science and finance, modeling population growth, radioactive decay, and compound interest.

D. $f(x) = -4x$

This is another linear function. As we discussed earlier, the domain of a linear function is always $(-\infty, \infty)$. The range of this particular linear function is also $(-\infty, \infty)$. As x increases or decreases without bound, so does $f(x)$. There is no upper or lower limit to the output values. Therefore, option D does not satisfy the required range of $(-\infty, 4]$. This further reinforces the principle that linear functions, while simple, have predictable and unbounded ranges.

Conclusion: The Correct Answer

After a thorough analysis of each option, we can confidently conclude that the correct answer is:

B. $f(x) = -x^2 + 4$

This function is the only one that satisfies both the domain requirement of $(-\infty, \infty)$ and the range requirement of $(-\infty, 4]$. The downward-opening parabola ensures that the function's output values are always less than or equal to its maximum value of 4.

Key Takeaways

This problem highlights the importance of understanding the definitions of domain and range and how they relate to different types of functions. By systematically analyzing each option, we were able to eliminate incorrect choices and identify the function that perfectly matched the given criteria. This approach can be applied to a wide range of function-related problems. Remember to:

• Define domain and range: Clearly understand what each term represents.
• Identify function type: Recognize the characteristics of linear, quadratic, exponential, and other function families.
• Analyze key features: Determine the vertex, intercepts, and asymptotic behavior of the function.
• Eliminate incorrect options: Rule out choices that do not meet the specified domain or range requirements.

By mastering these concepts and techniques, you will be well-equipped to tackle any function-related challenge. Understanding function behavior is a cornerstone of mathematical proficiency and opens doors to more advanced topics in calculus and beyond.

Additional Tips for Solving Domain and Range Problems

To further enhance your problem-solving skills, consider these additional tips:

• Graphing: Visualizing the function's graph can provide valuable insights into its domain and range. Use graphing calculators or online tools to plot the functions and observe their behavior.
• Transformations: Understand how transformations, such as shifts, stretches, and reflections, affect the domain and range of a function. For example, vertical shifts alter the range, while horizontal shifts affect the domain.
• Restricted Domains: Be mindful of functions with restricted domains, such as rational functions (where the denominator cannot be zero) and radical functions (where the radicand must be non-negative).
• Practice: The more problems you solve, the better you will become at identifying patterns and applying the concepts of domain and range. Seek out a variety of exercises and examples to challenge yourself.

By consistently applying these strategies, you will develop a strong intuition for domain and range and become a more confident and proficient problem-solver. Consistent practice is the key to mastering any mathematical concept.

This comprehensive guide has equipped you with the knowledge and tools necessary to confidently tackle domain and range problems. Remember to focus on understanding the fundamental concepts, analyzing the characteristics of different function types, and practicing consistently. With these skills, you will be well-prepared to excel in your mathematical endeavors.