The range of the function \(g(x)=|x-12|-2\) is \(\boxed{[-2,\infty)}\)
Further explanation:
Given:
The function is given as follows:
\(\boxed{g(x)=|x-12|-2}\)
The function \(g(x)\) is an absolute value function.
If \(x\leq12\) then the function \(g(x)\) can be written as follows:
\(\boxed{g(x)=-(x-12)-2}\)
Now, simplify the function \(g(x)=-(x-12)-2\) as follows:
\(\begin{aligned}g(x)&=-(x-12)-2\\&=-x+12-2\\&=-x+10\end{aligned}\)
For \(x\leq12\), the equation is
\(g(x)=-x+10\) …… (1)
Put \(x=12\) in the equation (1) to obtain the value of \(g(12)\).
\(\begin{aligned}g(12)&=-12+10\\&=-2\end{aligned}\)
Therefore, the minimum value of the function \(g(x)=-x+10\) is \(-2\).
If \(x\geq12\) then the function \(g(x)\) can be written as follows:
\(\boxed{g(x)=(x-12)-2}\)
Now, simplify the function \(g(x)=(x-12)-2\) as follows:
\(\begin{aligned}g(x)&=(x-12)-2\\&=x-12-2\\&=x-14\end{aligned}\)
For \(x\geq12\) the equation is,
\(g(x)=x-14\) …… (2)
Put \(x=12\) in equation (2) to obtain the value of \(g(12)\).
\(\begin{aligned}g(12)&=12-14\\&=-2\end{aligned}\)
Therefore, the minimum value of the function \(g(x)\) is \(-2\).
The minimum value of the function \(g(x)=|x-12|-2\) is \(-2\) and the maximum value is infinite.
The range of the function is the set of value of function at which the function is defined.
Thus the range of the function \(g(x)=|x-12|-2\) is \(\boxed{[-2,\infty)}\).
Learn more:
1. A problem on range of a function https://brainly.com/question/3412497
2. A problem on function https://brainly.com/question/2142762
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Function
Keywords: g(x)=|x-12|-2, range, g(x)=-(x-12)-2, g(x)=(x-12)-2, function, domain, absolute value of function, minimum value, maximum value, modulus, modulus function, range, domain.
