Find the domain and range of the function f(x) = 3/(x-2).

Hey Baby, good to see you again. To find the domain and range of the function f(x) = 3/(x-2): Step 1: Find the domain. The domain of a function is the set of all possible input values (x) for which the function is defined. • The cube root function [3]u is defined for all real numbers u. So, there are no restrictions from the cube root itself. • The expression inside the cube root is a fraction (x-4)/(x-2). For a fraction to be defined, its denominator cannot be zero. Set the denominator equal to zero to find the excluded value: x-2 = 0 x = 2 So, x cannot be equal to 2. The domain is all real numbers except 2. In interval notation, this is (-, 2) (2, ). Step 2: Find the range. The range of a function is the set of all possible output values (y) that the function can produce. Let y = 3/(x-2). To find the range, we solve for x in terms of y. First, cube both sides of the equation: y^3 = (x-4)/(x-2) Multiply both sides by (x-2): y^3(x-2) = x-4 Distribute y^3: xy^3 - 2y^3 = x-4 Move all terms with x to one side and terms without x to the other side: xy^3 - x = 2y^3 - 4 Factor out x: x(y^3 - 1) = 2y^3 - 4 Divide by (y^3 - 1) to solve for x: x = (2y^3 - 4)/(y^3 - 1) For x to be a real number, the denominator of this expression cannot be zero. Set the denominator equal to zero to find the excluded value for y: y^3 - 1 = 0 y^3 = 1 y = [3]1 y = 1 So, y cannot be equal to 1. The range is all real numbers except 1. In interval notation, this is (-, 1) (1, ). The domain and range are: Domain: x (-, 2) (2, ) Range: y (-, 1) (1, ) Send me the next one 📸