Sets, Relations and Functions

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Step 1: Form all ordered pairs where first element is from A and second from B. Step 2: (1,3), (1,4), (2,3), (2,4). Answer: A × B = {(1,3), (1,4), (2,3), (2,4)}…

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Step 1: Domain = set of first elements → {1,2,3}. Step 2: Range = set of second elements → {2,3,4}. Answer: Domain = {1,2,3}, Range = {2,3,4}…

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Step 1: Check if any input has more than one output. Step 2: Input '1' maps to both 2 and 4. Step 3: A function must have unique output for each input. Answer: No, because 1 has two images.

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Step 1: Denominator cannot be zero → x - 3 ≠ 0. Step 2: So x ≠ 3. Answer: Domain = R - {3}…

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Step 1: Expression under square root must be ≥ 0. Step 2: x - 5 ≥ 0 → x ≥ 5. Answer: Domain = [5, ∞)…

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Step 1: Denominator ≠ 0 → √(x+2) ≠ 0 → x ≠ -2. Step 2: Expression under root ≥ 0 → x + 2 ≥ 0 → x ≥ -2. Step 3: Combine: x > -2 (since x = -2 makes denominator 0). Answer: Domain = (-2, ∞)…

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Step 1: f(-1) = 2(-1)+1 = -1. Step 2: f(0) = 2(0)+1 = 1. Step 3: f(1) = 2(1)+1 = 3. Answer: Range = {-1, 1, 3}…

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Step 1: Minimum value at x = 0 → f(0) = 0. Step 2: Maximum at x = ±2 → f(2) = f(-2) = 4. Step 3: Since x² ≥ 0, range is from 0 to 4. Answer: Range = [0, 4]…