This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the domain and range for the functions: Example 1.2 (b): f(x) = sqrt(3x+2)-1 Step 1: Find the domain. For the function to be defined in real numbers, the expression under the square root must be non-negative. 3x+2 0 3x -2 x -(2)/(3) The domain is D(f) = [-(2)/(3), ). Step 2: Find the range. Since sqrt(3x+2) 0 for all x in the domain, we can determine the range of f(x). sqrt(3x+2)-1 0-1 f(x) -1 The range is R(f) = [-1, ). Final Answer for (b): Domain: [-(2)/(3), ) Range: [-1, ) Example 1.2 (c): f(x) = (x)/(x-2) Step 1: Find the domain. For a rational function, the denominator cannot be zero. x-2 0 x 2 The domain is D(f) = (-, 2) (2, ) or R \2\. Step 2: Find the range. Let y = f(x). We solve for x in terms of y. y = (x)/(x-2) y(x-2) = x yx - 2y = x yx - x = 2y x(y-1) = 2y x = (2y)/(y-1) For x to be defined, the denominator y-1 cannot be zero. y-1 0 y 1 The range is R(f) = (-, 1) (1, ) or R \1\. Final Answer for (c): Domain: (-, 2) (2, ) Range: (-, 1) (1, ) That's 2 down. 3 left today — send the next one.