Domain And Range Of F(x) = 2|x-4| Explained

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In mathematics, the domain and range of a function are fundamental concepts for understanding its behavior and characteristics. The domain refers to the set of all possible input values (x-values) for which the function is defined, while the range represents the set of all possible output values (f(x) or y-values) that the function can produce. Let's delve into determining the domain and range of the given absolute value function, f(x) = 2|x-4|, and explore the reasoning behind the correct answer.

Exploring the Domain of f(x) = 2|x-4|

To determine the domain of the function f(x) = 2|x-4|, we need to identify any restrictions on the input values, x. In other words, we need to find out if there are any values of x that would make the function undefined or result in an invalid output. Absolute value functions, in general, are defined for all real numbers. This is because the absolute value operation simply returns the non-negative magnitude of a number, regardless of whether the number is positive, negative, or zero. Consequently, there are no restrictions on the values of x that can be plugged into the expression |x-4|.

Furthermore, multiplying the absolute value expression by a constant, such as 2 in this case, does not introduce any new restrictions on the domain. The function remains defined for all real numbers. Therefore, the domain of f(x) = 2|x-4| is the set of all real numbers, which can be expressed in interval notation as (-∞, ∞). This signifies that any real number can be substituted for x in the function, and a corresponding output value will be produced.

In summary, the domain of f(x) = 2|x-4| is (-∞, ∞) because there are no values of x that would make the function undefined.

Unveiling the Range of f(x) = 2|x-4|

Now, let's turn our attention to determining the range of the function f(x) = 2|x-4|. The range represents the set of all possible output values (f(x) or y-values) that the function can produce. To find the range, we need to consider the behavior of the absolute value function and the effect of the constant multiplier.

The absolute value of any expression, |x-4| in this case, is always non-negative. This means that the output of the absolute value function will always be greater than or equal to zero. The minimum value of |x-4| is 0, which occurs when x = 4. As x moves away from 4 in either direction, the value of |x-4| increases.

Multiplying the absolute value expression by a positive constant, 2 in this case, scales the output values but does not change the non-negativity. The minimum value of 2|x-4| is 2 * 0 = 0, which still occurs when x = 4. As x moves away from 4, the value of 2|x-4| increases without bound. This means that the function can produce any non-negative value.

Therefore, the range of f(x) = 2|x-4| is the set of all non-negative real numbers, which can be expressed in interval notation as [0, ∞). Alternatively, we can express the range as f(x) ≥ 0, indicating that the output values are greater than or equal to zero.

In conclusion, the range of f(x) = 2|x-4| is [0, ∞) or f(x) ≥ 0 because the absolute value function always produces non-negative values, and the constant multiplier scales the output without changing the non-negativity.

Analyzing the Answer Choices

Now that we have determined the domain and range of f(x) = 2|x-4|, let's examine the given answer choices and identify the correct one:

  • A. domain: x ≤ 2; range: (-∞, ∞): This option is incorrect because the domain of the function is not restricted to x ≤ 2, and the range is not the set of all real numbers.
  • B. domain: (-∞, ∞), range: f(x) ≥ 0: This option is the correct answer. We have established that the domain of the function is indeed all real numbers, and the range is the set of all non-negative real numbers, represented by f(x) ≥ 0.
  • C. domain: (-∞, ∞), range: f(x) ≤ 0: This option is incorrect because the range of the function is not the set of all non-positive real numbers. The absolute value function ensures that the output values are always non-negative.
  • D. domain: x ≥ 2; range: (-∞, ∞): This option is incorrect because the domain of the function is not restricted to x ≥ 2, and the range is not the set of all real numbers.

Conclusion

In summary, the domain of the absolute value function f(x) = 2|x-4| is (-∞, ∞), and the range is f(x) ≥ 0. Option B accurately represents these findings. Understanding the domain and range of functions is crucial for comprehending their behavior and applications in various mathematical and real-world contexts. By analyzing the function's structure and properties, we can effectively determine the set of possible input and output values, providing valuable insights into its characteristics.

By mastering the concepts of domain and range, you gain a deeper understanding of functions and their role in mathematical analysis and problem-solving. So, continue to explore different types of functions, practice determining their domains and ranges, and unlock the power of mathematical understanding.

By understanding domain and range, we lay the groundwork for tackling more advanced mathematical concepts and real-world applications. Keep practicing, keep exploring, and keep expanding your mathematical horizons.