Question
What are the domain and range of \( f(x)=2|x-4| ? \) domain: \( x \leq 2 \); range: \( (-\infty, \infty) \) domain: \( (-\infty, \infty) \); range: \( f(x) \geq 0 \) domain: \( (-\infty, \infty) \); range: \( f(x) \leq 0 \) domain: \( x \geq 2 \); range: \( (-\infty, \infty) \)
Ask by Robbins Reeves. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Domain: All real numbers; Range: f(x) ≥ 0
Solution
To determine the domain and range of f(x) = 2|x – 4|, follow these steps:
1. Domain: The expression |x – 4| is defined for all real numbers (since absolute value is defined for any x). Multiplying by 2 doesn't restrict the domain further. So the domain is all real numbers, i.e., (–∞, ∞).
2. Range: The absolute value |x – 4| is always nonnegative (≥ 0), and multiplying it by 2 still yields nonnegative values. Hence, f(x) is always ≥ 0, meaning the range is [0, ∞).
Comparing with the given options, the correct answer is:
domain: (–∞, ∞); range: f(x) ≥ 0.
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Simplify this solution Extra Insights
The function \( f(x) = 2|x - 4| \) has a domain of all real numbers, which means you can input any value of \( x \) anywhere on the number line. So, the domain is \( (-\infty, \infty) \). The range relates to the output of the function; since the absolute value is always non-negative and multiplied by 2, the minimum output is 0, leading to a range of \( [0, \infty) \). To visualize this function, we can think about its "V" shape centered at the point (4, 0). This shape emerges from the absolute value function, and every output value will be above or equal to 0. So, when considering different values of \( x \), you'll see that you can't get any negative outputs! Isn't that neat?
