Question

Determine the domain of the functions (a) f(x)= x−3 ​ and (b) g(x)= x−3 ​ 1 ​ . Write your answers in interval notation. Give an example of a function whose domain does not include x=−3 or x=3. Give an example of a function whose domain is [−3,3]. Give an example of a function whose domain is (−3,3).

Answer 1

The domain of f(x) = x - 3 is (-∞, ∞) and the domain of g(x) = (x - 3)/(1 - x) is (-∞, 1)∪(1, ∞). Examples of functions with different domains are provided.

Step-by-step explanation:

The domain of a function refers to the set of all possible input values, or x-values, for which the function is defined. For the function f(x) = x - 3, the domain is all real numbers since there are no restrictions on x.

In interval notation, the domain is (-∞, ∞). For the function g(x) = (x - 3)/(1 - x), the domain is all real numbers except x = 1 because it would result in division by zero. In interval notation, the domain is (-∞, 1)∪(1, ∞).

An example of a function whose domain does not include x = -3 or x = 3 is h(x) = √(x - 2).

The square root function is not defined for negative numbers, so the domain of h(x) is x ≥ 2.

An example of a function with a domain of [-3, 3] is f(x) = x^2, since x^2 is defined for all values of x in that interval.

An example of a function with a domain of (-3, 3) is g(x) = 1/x, where x ≠ 0.

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