What is the domain of the function [tex]$f(x)=3|x+4|+1$[/tex]?
A. all real numbers
B. all real numbers less than or equal to -4
C. all real numbers greater than or equal to 1
D. all real numbers greater than or equal to -4
Answer (2)
The function is f ( x ) = 3∣ x + 4∣ + 1 .
The absolute value function ∣ x + 4∣ is defined for all real numbers.
Multiplying by 3 and adding 1 does not change the domain.
The domain of f ( x ) is all real numbers. all real numbers
Explanation
Understanding the Problem We are asked to find the domain of the function f ( x ) = 3∣ x + 4∣ + 1 . The domain of a function is the set of all possible values of x for which the function is defined.
Analyzing the Absolute Value The absolute value function ∣ x + 4∣ is defined for all real numbers x . This is because for any real number x , we can compute x + 4 , and then take the absolute value of the result.
Multiplication by a Constant Since ∣ x + 4∣ is defined for all real numbers, 3∣ x + 4∣ is also defined for all real numbers. Multiplying by a constant does not change the domain.
Adding a Constant Adding 1 to 3∣ x + 4∣ does not change the domain either, since we can add 1 to any real number. Therefore, f ( x ) = 3∣ x + 4∣ + 1 is defined for all real numbers.
Final Answer The domain of f ( x ) = 3∣ x + 4∣ + 1 is all real numbers.
Examples
Understanding the domain of a function is crucial in many real-world applications. For example, if x represents the number of items produced in a factory, the function f ( x ) might represent the cost of production. Since you can't produce a negative number of items, the domain would be all non-negative real numbers. Similarly, if x represents time, and f ( x ) represents the amount of water in a tank, the domain would be restricted to the time intervals when the tank is being filled or emptied. In this case, the absolute value function ensures that the result is always non-negative, which can be useful in modeling situations where only the magnitude of a quantity matters, such as distance or speed.
The domain of the function f ( x ) = 3∣ x + 4∣ + 1 is all real numbers. This is because the absolute value function and subsequent operations do not restrict the values of x . Therefore, the correct answer is option A: all real numbers.
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