What is the range of the function [tex]g(x)=|x-12|-2[/tex]?
[tex] \{y \mid y\ \textgreater \ -2\} [/tex]
[tex] \{y \mid y \geq-2\} [/tex]
[tex] \{y \mid y\ \textgreater \ 12\} [/tex]
[tex] \{y \mid y \geq 12\} [/tex]
Answer (1)
The function is g ( x ) = ∣ x − 12∣ − 2 .
The minimum value of ∣ x − 12∣ is 0, which occurs at x = 12 .
The minimum value of g ( x ) is 0 − 2 = − 2 .
The range of g ( x ) is {y \mid y \geq -2\} , so the answer is y ∣ y ≥ − 2 .
Explanation
Understanding the Problem We are given the function g ( x ) = ∣ x − 12∣ − 2 and asked to find its range. 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 − 12∣ is always non-negative, meaning its minimum value is 0. This occurs when x = 12 . Therefore, the smallest value of ∣ x − 12∣ is 0.
Finding the Minimum Value of g(x) Since g ( x ) = ∣ x − 12∣ − 2 , the minimum value of g ( x ) occurs when ∣ x − 12∣ is at its minimum, which is 0. So, the minimum value of g ( x ) is 0 − 2 = − 2 .
Determining the Range As x moves away from 12, the value of ∣ x − 12∣ increases. Since ∣ x − 12∣ can take any non-negative value, g ( x ) can take any value greater than or equal to -2. Therefore, the range of g ( x ) is all real numbers greater than or equal to -2.
Expressing the Range in Set Notation In set notation, the range of g ( x ) is written as y { y ≥ − 2 } .
Examples
Understanding the range of absolute value functions is useful in various real-world scenarios. For example, consider a thermostat set to 20 degrees Celsius. The function f ( t ) = ∣ t − 20∣ represents the difference between the actual temperature t and the set temperature. If the thermostat has a tolerance of ± 2 degrees, the range of acceptable temperatures is [ 18 , 22 ] . Similarly, in manufacturing, absolute value functions can model deviations from a target measurement, and understanding the range helps determine acceptable tolerance levels.
