What are the domain and range of $f(x)=|x+6|$?
A. domain: $(-\infty, \infty)$; range: $f(x) \geq 0$
B. domain: $x \leq-6 ; range: (-\infty, \infty)$
C. domain: $x \geq-6$; range: $(-\infty, \infty)$
D. domain: $(-\infty, \infty)$; range: $f(x) \leq 0$
Answer (1)
The domain of the function f ( x ) = ∣ x + 6∣ is all real numbers.
The range of the function f ( x ) = ∣ x + 6∣ is all non-negative real numbers.
The domain is ( − ∞ , ∞ ) .
The range is f ( x ) ≥ 0 .
domain: ( − ∞ , ∞ ) ; range: f ( x ) ≥ 0
Explanation
Understanding the Problem We are asked to find the domain and range of the function f ( x ) = ∣ x + 6∣ . Let's break this down.
Determining the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, f ( x ) = ∣ x + 6∣ is an absolute value function. We can plug any real number into the absolute value function, so the domain is all real numbers.
Determining the Range The range of a function is the set of all possible output values (f(x)-values). The absolute value function always returns a non-negative value. The minimum value of ∣ x + 6∣ is 0, which occurs when x = − 6 . Therefore, the range is all non-negative real numbers.
Expressing the Domain and Range In interval notation, the domain is ( − ∞ , ∞ ) and the range is [ 0 , ∞ ) . The range can also be written as f ( x ) ≥ 0 .
Examples
Imagine you're designing a symmetrical garden bed around a central point. The function f(x) = |x+6| can represent the distance from the center, where x is the position relative to a reference point. The domain (-∞, ∞) means you can place plants anywhere along the line, and the range f(x) ≥ 0 ensures that the distance is always non-negative, making the garden symmetrical and physically possible. This concept is useful in various symmetrical designs, ensuring measurements are always meaningful.
