Question

Find the and range for both the function ans inverse. f(x)=(2)/(3)x-5

Answer 1

The domain of the function f(x) = 2/3x - 5 is all real numbers
`\(\mathbb{R}\)`. The range is also all real numbers
`\(\mathbb{R}\)`. For the inverse function
`\(f^(-1)(x)\)`, the domain and range are reversed: the domain is
`\(\mathbb{R}\)`, and the range is all real numbers
`\(\mathbb{R}\)`.

Step-by-step explanation:

The given function is a linear function f(x) = 2/3x - 5 . The domain of a linear function is all real numbers, meaning that any real number can be plugged into the function. Therefore, the domain of f(x) is
`\(\mathbb{R}\)`, indicating that the function is defined for all real values of (x).

The range of a linear function is also all real numbers unless the function is horizontal, which is not the case here. In this instance, the function f(x) is not constrained, and as (x) varies over all real numbers, the output f(x) spans all real values. Therefore, the range of f(x) is
`\(\mathbb{R}\)`, indicating that the function can produce any real number.

For the inverse function
`\(f^(-1)(x)\)`, the roles of domain and range are swapped. The domain of
`\(f^(-1)(x)\)`is determined by the range of f(x), and vice versa. Since the range of f(x) is
`\(\mathbb{R}\)`, the domain of
`\(f^(-1)(x)\)` is also
`\(\mathbb{R}\)` Similarly, as the domain of f(x) is
`\(\mathbb{R}\)`, the range of
`\(f^(-1)(x)\)` encompasses all real numbers
`\(\mathbb{R}\)`.