Range Of F(x) = -4/x + 1 Calculation And Explanation

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In the realm of mathematics, understanding the behavior of functions is paramount. One crucial aspect of this understanding is determining the range of a function. The range represents the set of all possible output values that a function can produce. In this article, we will delve deep into finding the range of the function f(x) = -4/x + 1. This exploration will not only provide the answer but also elucidate the underlying concepts and techniques involved in range determination.

The function in question, f(x) = -4/x + 1, is a rational function. Rational functions are functions that can be expressed as the ratio of two polynomials. In this case, the numerator is the constant -4, and the denominator is the simple polynomial x. The addition of 1 to the term -4/x introduces a vertical shift, which will play a significant role in shaping the function's range. To effectively determine the range, we need to analyze how the function behaves as x takes on different values.

Analyzing the Behavior of f(x) = -4/x + 1

To truly grasp the range of f(x) = -4/x + 1, we need to examine its behavior across the entire spectrum of possible x-values. This includes considering both positive and negative values, as well as values approaching zero and infinity. Let's break down this analysis step by step:

1. The Impact of -4/x

The core of our function lies in the term -4/x. This term exhibits a crucial characteristic: as x gets closer to zero, the absolute value of -4/x becomes increasingly large. However, the sign of -4/x depends on the sign of x itself. When x is positive, -4/x is negative, and as x approaches zero from the positive side, -4/x plunges towards negative infinity. Conversely, when x is negative, -4/x is positive, and as x approaches zero from the negative side, -4/x soars towards positive infinity. This behavior introduces a vertical asymptote at x = 0, a critical point in our range analysis. The presence of this asymptote means the function will never actually equal a value at x=0, thus affecting the range.

2. The Role of +1: Vertical Shift

The addition of 1 to the term -4/x introduces a vertical shift to the function's graph. This means that the entire graph of -4/x is shifted upwards by one unit. Consequently, the horizontal axis, which would normally be a horizontal asymptote for -4/x, is also shifted upwards by one unit. This creates a new horizontal asymptote at y = 1. The horizontal asymptote provides another key piece of information about the range, indicating a value that the function approaches but never actually reaches.

3. Considering Extreme Values of x

To complete our analysis, we must also consider what happens to f(x) as x moves towards positive and negative infinity. As x becomes extremely large (either positively or negatively), the term -4/x approaches zero. This is because dividing a constant by an increasingly large number results in a value that gets closer and closer to zero. Therefore, as x approaches infinity, f(x) approaches 0 + 1 = 1. This observation reinforces the presence of the horizontal asymptote at y = 1.

Determining the Range

Having analyzed the behavior of f(x) = -4/x + 1, we are now equipped to determine its range. We've established the following key points:

  • There is a vertical asymptote at x = 0, meaning the function is undefined at x = 0.
  • There is a horizontal asymptote at y = 1, meaning the function approaches but never equals 1.
  • As x approaches positive infinity, f(x) approaches 1 from below.
  • As x approaches negative infinity, f(x) approaches 1 from above.
  • As x approaches 0 from the positive side, f(x) approaches negative infinity.
  • As x approaches 0 from the negative side, f(x) approaches positive infinity.

Combining these insights, we can conclude that the function f(x) = -4/x + 1 can take on any real value except for 1. It covers all values below 1 as x approaches positive infinity and 0 from the positive side, and it covers all values above 1 as x approaches negative infinity and 0 from the negative side. Therefore, the range of f(x) = -4/x + 1 is the set of all real numbers except 1.

Expressing the Range in Interval Notation

To express the range concisely, we can use interval notation. In interval notation, we use parentheses and brackets to indicate whether endpoints are included in the interval. Parentheses indicate that an endpoint is not included, while brackets indicate that it is. Since the function f(x) = -4/x + 1 can take on any value except 1, we express its range as the union of two intervals:

(-\infty, 1) ∪ (1, \infty)

This notation signifies that the range includes all real numbers less than 1 and all real numbers greater than 1, excluding 1 itself. This is a standard way to represent ranges that have a discontinuity or a value that is not included.

Why Option C is the Correct Answer

Given the analysis above, we can confidently identify the correct answer to the question "What is the range of f(x) = -4/x + 1?". The options provided were:

A. (-∞, 0) ∪(0, ∞) B. (-∞,-1) ∪(-1, ∞) C. (-∞, 1) ∪(1, ∞) D. (-∞,-4) ∪(-4, ∞)

As we have demonstrated through our analysis, the correct answer is C. (-∞, 1) ∪(1, ∞). This option accurately represents the range of the function f(x) = -4/x + 1, which includes all real numbers except 1.

Key Concepts and Techniques

The process of determining the range of f(x) = -4/x + 1 highlights several key concepts and techniques that are essential in the study of functions:

  • Rational Functions: Understanding the behavior of rational functions, particularly the role of asymptotes, is crucial for determining their ranges. Vertical asymptotes arise from values of x that make the denominator zero, while horizontal asymptotes are determined by the behavior of the function as x approaches infinity.
  • Vertical Shifts: Recognizing the effect of vertical shifts on the graph of a function is essential. Adding a constant to a function shifts its graph vertically, affecting the range by shifting the horizontal asymptote as well.
  • Limit Analysis: Evaluating the limits of a function as x approaches specific values, including infinity and points of discontinuity, provides valuable information about the function's behavior and its range. In our case, analyzing the limits as x approached 0 and infinity helped us identify the asymptotes and understand how the function behaves in extreme cases.
  • Interval Notation: Mastering interval notation is crucial for expressing ranges and other sets of numbers concisely and accurately. The use of parentheses and brackets allows us to clearly indicate whether endpoints are included or excluded from the interval.

Conclusion

Determining the range of a function is a fundamental skill in mathematics. By carefully analyzing the behavior of f(x) = -4/x + 1, we have successfully identified its range as (-∞, 1) ∪ (1, ∞). This process involved understanding the roles of vertical and horizontal asymptotes, vertical shifts, and limit analysis. The key takeaway is that a comprehensive analysis of a function's behavior, combined with the appropriate mathematical tools, allows us to accurately determine its range. This understanding is not only valuable for solving specific problems but also for developing a deeper appreciation for the nature of functions and their properties. This exploration hopefully provided a solid understanding and the techniques to apply to finding the range of other functions.

By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of problems involving functions and their ranges. Remember to always analyze the behavior of the function, consider asymptotes and shifts, and use interval notation to express your results clearly and concisely.