What are the domain and range of [tex]f(x)=|x-3|+6[/tex]?
A. Domain: [tex] \{x \mid x \text{ is all real numbers }\} [/tex]
Range: [tex] \{y \mid y \geq 6\} [/tex]
B. Domain: [tex] \{x \mid x \geq 3\} [/tex]
Range: [tex] \{y \mid y \geq 6\} [/tex]
C. Domain: [tex] \{x \mid x \text{ is all real numbers }\} [/tex]
Range: [tex] \{y \mid y \geq-6\} [/tex]
D. Domain: [tex] \{x \mid x \geq 3\} [/tex]
Range: [tex] \{y \mid y \geq-6\} [/tex]
Answer (1)
The domain of f ( x ) = ∣ x − 3∣ + 6 is all real numbers.
The absolute value ∣ x − 3∣ is always non-negative, so ∣ x − 3∣ ≥ 0 .
Adding 6 to both sides gives f ( x ) = ∣ x − 3∣ + 6 ≥ 6 .
Therefore, the range is y ≥ 6 . The final answer is Domain: x ∣ x is all real numbers , Range: y ∣ y ≥ 6 .
Explanation
Understanding the Function The function given is f ( x ) = ∣ x − 3∣ + 6 . We need to find its domain and range.
Determining the Domain The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since the absolute value function is defined for all real numbers, there are no restrictions on the values of x that can be plugged into f ( x ) . Therefore, the domain of f ( x ) is all real numbers.
Analyzing the Range The range of a function is the set of all possible output values (y-values) that the function can produce. The absolute value ∣ x − 3∣ is always non-negative, meaning ∣ x − 3∣ ≥ 0 for all x .
Finding the Range Adding 6 to both sides of the inequality, we get ∣ x − 3∣ + 6 ≥ 0 + 6 , which simplifies to f ( x ) ≥ 6 . This means that the smallest possible value of f ( x ) is 6. Since the absolute value can be arbitrarily large, f ( x ) can take any value greater than or equal to 6. Therefore, the range of f ( x ) is all real numbers greater than or equal to 6.
Final Answer In conclusion, the domain of f ( x ) = ∣ x − 3∣ + 6 is all real numbers, and the range is y ≥ 6 .
Examples
Imagine you're tracking the temperature change relative to a baseline of 6 degrees Celsius. The function f ( x ) = ∣ x − 3∣ + 6 models this, where x is the actual temperature. The domain being all real numbers means you can input any temperature. The range y ≥ 6 tells you the temperature will always be at least 6 degrees, showing how absolute value functions ensure values stay above a certain level, useful in scenarios like setting minimum safety thresholds or analyzing deviations from a standard.
