Final answer:
The domain of the function f(x) = |x| - 7.5 is all real numbers, and the range is all real numbers greater than or equal to -7.5. These are expressed in set notation as x for the domain and f(x) for the range, and in interval notation as (-∞, ∞) for the domain and [-7.5, ∞) for the range.
Step-by-step explanation:
To answer the question regarding the function f(x) = |x| - 7.5, we need to establish its domain and range in both set notation and interval notation.
Domain of f(x) = |x| - 7.5
The domain of a function is the set of all possible input values (x-values) for which the function is defined. Since absolute value is defined for all real numbers, and no other constraints are given, the domain of f(x) is all real numbers.
Set Notation: The domain is x , where ℝ represents the set of all real numbers.
Interval Notation: The domain is (-∞, ∞).
Range of f(x) = |x| - 7.5
The range of a function is the set of all possible output values (y-values). This function takes the absolute value of x and then subtracts 7.5, which means the smallest value it can take is -7.5 (when x = 0). As x increases in either positive or negative direction, |x| increases, which means f(x) will also increase. Therefore, the range is all real numbers greater than or equal to -7.5.
Set Notation: The range is f(x) .
Interval Notation: The range is [-7.5, ∞).