The function $g(x)=|x-6|-8$ is graphed. What is the range?
A. {y | y>-8}
B. {y | y \geq-8}
C. {y | y<-8}
D. {y | y is all real numbers }
Answer (1)
The absolute value function ∣ x − 6∣ is always non-negative.
Therefore, g ( x ) = ∣ x − 6∣ − 8 is always greater than or equal to − 8 .
The minimum value of g ( x ) is − 8 , which occurs when x = 6 .
The range of g ( x ) is y ∣ y ≥ − 8 .
Explanation
Understanding the Problem The problem asks us to find the range of the function g ( x ) = ∣ x − 6∣ − 8 . The range of a function is the set of all possible output values (y-values) that the function can produce.
Analyzing the Absolute Value The absolute value function ∣ x − 6∣ is always non-negative, meaning it is greater than or equal to zero for any value of x . That is, ∣ x − 6∣ ≥ 0 for all x . The smallest value of ∣ x − 6∣ is 0, which occurs when x = 6 .
Finding the Minimum Value Since ∣ x − 6∣ ≥ 0 , we can say that g ( x ) = ∣ x − 6∣ − 8 ≥ 0 − 8 = − 8 . This tells us that the smallest possible value for g ( x ) is − 8 . This occurs when ∣ x − 6∣ = 0 , which happens when x = 6 . Thus, g ( 6 ) = ∣6 − 6∣ − 8 = 0 − 8 = − 8 .
Determining the Range As x moves away from 6, the value of ∣ x − 6∣ increases, and therefore, the value of g ( x ) = ∣ x − 6∣ − 8 also increases. Since ∣ x − 6∣ can take any non-negative value, g ( x ) can take any value greater than or equal to − 8 . Therefore, the range of g ( x ) is all y such that y ≥ − 8 .
Final Answer The range of the function g ( x ) = ∣ x − 6∣ − 8 is the set of all y values such that y is greater than or equal to − 8 . In set notation, this is written as y ∣ y ≥ − 8 .
Examples
Imagine you are designing a temperature control system where the temperature is represented by the function g ( x ) = ∣ x − 6∣ − 8 , where x is a control variable. The range tells you the possible temperature values the system can achieve. Knowing that the temperature will always be greater than or equal to -8 degrees is crucial for designing a functional and safe system. This ensures that you can predict and manage the temperature within acceptable limits, preventing it from dropping below a critical threshold.
