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
Step 1: Understand the function and the given range. The function is f(y) = 1 - (3)/(y). The range is given as the interval (-2, 7), which means -2 < f(y) < 7. We need to find the values of y (the domain) for which f(y) falls within this range. Step 2: Set up inequalities based on the range. We have two inequalities to solve: *a) f(y) > -2 *b) f(y) < 7 Step 3: Solve inequality a) for y. 1 - (3)/(y) > -2 Add 2 to both sides: 3 - (3)/(y) > 0 Multiply by y (we must consider two cases for y > 0 and y < 0): Case 1: y > 0 3y - 3 > 0 3y > 3 y > 1 So, for y > 0, the condition is y > 1. Case 2: y < 0 When multiplying by a negative number, reverse the inequality sign: 3y - 3 < 0 3y < 3 y < 1 So, for y < 0, the condition is y < 1. Since we assumed y < 0, this means all y < 0 satisfy the inequality. Combining both cases for f(y) > -2, the solution is y < 0 or y > 1. In interval notation: (-, 0) (1, ). Step 4: Solve inequality b) for y. 1 - (3)/(y) < 7 Subtract 1 from both sides: -(3)/(y) < 6 Multiply by -1 and reverse the inequality sign: (3)/(y) > -6 Multiply by y (again, consider two cases): Case 1: y > 0 3 > -6y -(3)/(6) < y y > -(1)/(2) So, for y > 0, the condition is y > -(1)/(2). Since we assumed y > 0, this means all y > 0 satisfy the inequality. Case 2: y < 0 When multiplying by a negative number, reverse the inequality sign: 3 < -6y -(3)/(6) > y y < -(1)/(2) So, for y < 0, the condition is y < -(1)/(2). Combining both cases for f(y) < 7, the solution is y < -(1)/(2) or y > 0. In interval notation: (-, -(1)/(2)) (0, ). Step 5: Find the intersection of the solutions from Step 3 and Step 4. We need y to satisfy both conditions: Condition 1: y (-, 0) (1, ) Condition 2: y (-, -(1)/(2)) (0, ) Let's find the intersection of these two sets: The intersection of (-, 0) and (-, -(1)/(2)) is (-, -(1)/(2)). The intersection of (1, ) and (0, ) is (1, ). The other intersections are empty. Therefore, the domain for the given range is y < -(1)/(2) or y > 1. In interval notation, the domain is (-, -(1)/(2)) (1, ). Step 6: Present the answer on an arrow diagram. An arrow diagram shows the mapping from the domain to the range. Domain: D = \y R y < -(1)/(2) or y > 1\ Range: R = \f(y) R -2 < f(y) < 7\ The arrow diagram can be represented as: Domain (-, -(1)/(2)) (1, ) f(y) = 1 - (3)/(y) Range (-2, 7)