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.
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To find the domain and range of the function f(x) = (x-1)/(|x|): 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 numerator x-1 is defined for all real numbers. • The denominator |x| cannot be zero, because division by zero is undefined. Set the denominator equal to zero to find the excluded value: |x| = 0 x = 0 So, x cannot be equal to 0. The domain is all real numbers except 0. In interval notation, this is (-, 0) (0, ). 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 = (x-1)/(|x|). We need to consider two cases for x: Case 1: x > 0 If x > 0, then |x| = x. The function becomes: y = (x-1)/(x) y = 1 - (1)/(x) As x takes values greater than 0: • As x 0^+ (approaches 0 from the positive side), (1)/(x) , so y = 1 - -. • As x , (1)/(x) 0, so y = 1 - 0 1. For x > 0, the values of y are in the interval (-, 1). Case 2: x < 0 If x < 0, then |x| = -x. The function becomes: y = (x-1)/(-x) y = -((x-1)/(x)) y = -(1 - (1)/(x)) y = -1 + (1)/(x) As x takes values less than 0: • As x 0^- (approaches 0 from the negative side), (1)/(x) -, so y = -1 + (-) -. • As x -, (1)/(x) 0, so y = -1 + 0 -1. For x < 0, the values of y are in the interval (-, -1). Step 3: Combine the ranges from both cases. The total range is the union of the ranges from Case 1 and Case 2: Range = (-, 1) (-, -1) The union of these two intervals is (-, 1). The domain and range are: Domain: x (-, 0) (0, ) Range: y (-, 1) That's 2 down. 3 left today — send the next one.