The range of the function [tex]g(x)=|x-12|-2[/tex] is [tex]\boxed{[-2,\infty)}[/tex]
Further explanation:
Given:
The function is given as follows:
[tex]\boxed{g(x)=|x-12|-2}[/tex]
The function [tex]g(x)[/tex] is an absolute value function.
If [tex]x\leq12[/tex] then the function [tex]g(x)[/tex] can be written as follows:
[tex]\boxed{g(x)=-(x-12)-2}[/tex]
Now, simplify the function [tex]g(x)=-(x-12)-2[/tex] as follows:
[tex]\begin{aligned}g(x)&=-(x-12)-2\\&=-x+12-2\\&=-x+10\end{aligned}[/tex]
For [tex]x\leq12[/tex], the equation is
[tex]g(x)=-x+10[/tex] …… (1)
Put [tex]x=12[/tex] in the equation (1) to obtain the value of [tex]g(12)[/tex].
[tex]\begin{aligned}g(12)&=-12+10\\&=-2\end{aligned}[/tex]
Therefore, the minimum value of the function [tex]g(x)=-x+10[/tex] is [tex]-2[/tex].
If [tex]x\geq12[/tex] then the function [tex]g(x)[/tex] can be written as follows:
[tex]\boxed{g(x)=(x-12)-2}[/tex]
Now, simplify the function [tex]g(x)=(x-12)-2[/tex] as follows:
[tex]\begin{aligned}g(x)&=(x-12)-2\\&=x-12-2\\&=x-14\end{aligned}[/tex]
For [tex]x\geq12[/tex] the equation is,
[tex]g(x)=x-14[/tex] …… (2)
Put [tex]x=12[/tex] in equation (2) to obtain the value of [tex]g(12)[/tex].
[tex]\begin{aligned}g(12)&=12-14\\&=-2\end{aligned}[/tex]
Therefore, the minimum value of the function [tex]g(x)[/tex] is [tex]-2[/tex].
The minimum value of the function [tex]g(x)=|x-12|-2[/tex] is [tex]-2[/tex] 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 [tex]g(x)=|x-12|-2[/tex] is [tex]\boxed{[-2,\infty)}[/tex].
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.
