Finding The Range Of F(x) = -4|x+1| - 5 A Step-by-Step Guide
In this comprehensive guide, we will delve into the process of determining the range of the function f(x) = -4|x+1| - 5. Understanding the range of a function is crucial in mathematics as it defines the set of all possible output values the function can produce. This article will provide a step-by-step explanation, making it easier to grasp the concepts involved. We will break down the function, analyze its components, and discuss the impact of each part on the final range. By the end of this guide, you will have a solid understanding of how to find the range of this particular function and similar functions.
The question we aim to solve is: What is the range of the function f(x) = -4|x+1| - 5?
Let's consider the function f(x) = -4|x+1| - 5. To find the range, we need to determine all possible values that f(x) can take. This involves understanding the behavior of the absolute value function and how transformations affect it. The absolute value function, denoted as |x|, always returns a non-negative value, meaning |x| is always greater than or equal to zero. This property is fundamental to understanding the range of our given function. When we introduce transformations such as multiplication by a negative number and vertical shifts, the range changes accordingly. The absolute value part, |x+1|, ensures the expression inside the absolute value is considered in terms of its magnitude, not its sign. This means that whether x+1 is positive or negative, the result of |x+1| will always be non-negative. The multiplication by -4 flips the non-negative values to non-positive values, and the subtraction of 5 further shifts the range downwards. By analyzing these transformations step-by-step, we can accurately determine the range of the function.
Breaking Down the Function
1. The Absolute Value Component: |x+1|
First, let's analyze the absolute value part, |x+1|. The absolute value function always returns a non-negative value, meaning that |x+1| ≥ 0 for all real numbers x. The expression inside the absolute value, x+1, shifts the standard absolute value function one unit to the left. However, this shift does not affect the range, which remains non-negative. The minimum value of |x+1| is 0, which occurs when x = -1. As x moves away from -1 in either direction, the value of |x+1| increases. Therefore, the output of |x+1| can be any non-negative real number.
2. Multiplication by -4: -4|x+1|
Next, consider the effect of multiplying the absolute value by -4, resulting in -4|x+1|. Multiplying by a negative number reflects the function across the x-axis and also stretches it vertically. Since |x+1| is always non-negative, multiplying it by -4 makes the entire expression non-positive. In other words, -4|x+1| ≤ 0. The maximum value of -4|x+1| is 0, which occurs when |x+1| = 0, i.e., when x = -1. For any other value of x, -4|x+1| will be negative. This transformation significantly alters the range, making it non-positive instead of non-negative.
3. Vertical Shift by -5: -4|x+1| - 5
Finally, we subtract 5 from the expression, obtaining f(x) = -4|x+1| - 5. Subtracting 5 from the expression shifts the entire function downwards by 5 units. This vertical shift affects the range by reducing all values by 5. Since the maximum value of -4|x+1| is 0, the maximum value of f(x) = -4|x+1| - 5 is 0 - 5 = -5. As -4|x+1| can take any non-positive value, f(x) can take any value less than or equal to -5. Therefore, the range of the function is all real numbers less than or equal to -5.
Determining the Range
Combining these transformations, we can clearly see how the range of the function f(x) = -4|x+1| - 5 is determined. The absolute value part, |x+1|, gives us non-negative values. Multiplying by -4 flips these values to non-positive, and subtracting 5 shifts the entire function down by 5 units. Consequently, the maximum value of the function is -5, and it extends downwards to negative infinity. This understanding is crucial for selecting the correct answer from the given options.
To summarize the process:
- |x+1| has a range of [0, ∞).
- -4|x+1| has a range of (-∞, 0].
- -4|x+1| - 5 has a range of (-∞, -5].
Therefore, the range of the function f(x) = -4|x+1| - 5 is (-∞, -5]. This means that the function can take any value from negative infinity up to and including -5.
Conclusion
In conclusion, by carefully analyzing each component of the function f(x) = -4|x+1| - 5, we have determined that the range is (-∞, -5]. This involves understanding the properties of the absolute value function, the effect of multiplication by a negative number, and the impact of vertical shifts. The step-by-step approach allows us to break down the problem and understand how each transformation contributes to the final range. The correct answer is A. (-∞, -5].
Understanding the range of functions is a fundamental concept in mathematics, and this detailed explanation should provide a clear understanding of how to approach similar problems in the future. By breaking down complex functions into simpler parts and analyzing the impact of each transformation, we can accurately determine the range and gain a deeper understanding of the function's behavior. Remember to always consider the properties of the basic functions and how transformations alter their characteristics.
