Final answer:
The domain of the function f(x)=2-|x-5| is all real numbers.
Step-by-step explanation:
The domain of the function f(x)=2-|x-5| is all real numbers. In this function, the expression |x-5| represents the absolute value of (x-5). The absolute value is always non-negative, so it can be any real number or zero. Therefore, there are no restrictions on the values that x can take, and the domain is all real numbers.
In mathematics, the domain of a function is the set of all possible inputs (values of x) for which the function is defined. To find the domain of the given function f(x) = 2 - |x - 5|, we must consider the nature of the absolute value function.
The absolute value function |x - 5| is defined for all real numbers because absolute value simply provides the distance of a number from 5 on the real number line, regardless of whether the input is a positive number, a negative number, or zero. In general, for any real number a, the function |a| is defined.
In our function, a is replaced by (x - 5), which also means that the absolute value |x - 5| is defined for all real numbers x. When we consider the rest of the function f(x) = 2 - |x - 5|, there is nothing that restricts the domain further.
Subtracting the absolute value of a real number from 2 does not impose any limitations on the values that x can take.
Therefore, the domain of the function f(x) is all real numbers.
In interval notation, we write this as (-∞, ∞), where "∞" represents infinity and the parentheses indicate that infinity is not a real number and is not included in the domain, but rather suggests that there is no upper or lower bound to the set of possible values of x. In conclusion, the domain of the function f(x) = 2 - |x - 5| is the set of all real numbers.