Domain and Range

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Answer Key & Explanations

Domain and Range · Grade 9 · Worksheet 2

1. f(x) = 2x² - 7x + 3; f(2) = ? Answer: -3 Solution: Start with the function f(x) = 2x² - 7x + 3 Substitute x = 2 into the function: f(2) = 2(2)² - 7(2) + 3 Calculate the exponent first: (2)² = 4 Multiply: 2 × 4 = 8 and -7 × 2 = -14 Combine all terms: 8 - 14 + 3 Perform the operations from left to right: 8 - 14 = -6, then -6 + 3 = -3 The answer is -3.
   Full step-by-step solution

   Step 1: Start with the function f(x) = 2x² - 7x + 3 Step 2: Substitute x = 2 into the function: f(2) = 2(2)² - 7(2) + 3 Step 3: Calculate the exponent first: (2)² = 4 Step 4: Multiply: 2 × 4 = 8 and -7 × 2 = -14 Step 5: Combine all terms: 8 - 14 + 3 Step 6: Perform the operations from left to right: 8 - 14 = -6, then -6 + 3 = -3 The answer is -3.

2. f(x) = 2x² - 5x + 3; f(4) = ? Answer: 15 Solution: Write the function: f(x) = 2x² - 5x + 3 Substitute x = 4: f(4) = 2(4)² - 5(4) + 3 Calculate the exponent: (4)² = 16 Multiply: 2 × 16 = 32 and -5 × 4 = -20 Add all terms: 32 - 20 + 3 32 - 20 = 12, then 12 + 3 = 15 The answer is 15.
   Full step-by-step solution

   Step 1: Write the function: f(x) = 2x² - 5x + 3 Step 2: Substitute x = 4: f(4) = 2(4)² - 5(4) + 3 Step 3: Calculate the exponent: (4)² = 16 Step 4: Multiply: 2 × 16 = 32 and -5 × 4 = -20 Step 5: Add all terms: 32 - 20 + 3 Step 6: 32 - 20 = 12, then 12 + 3 = 15 The answer is 15.

3. A quadratic function is graphed on a coordinate plane with its vertex at (2, -3) and passing through the point (4, 1). The parabola opens upward. What is the equation of this quadratic function in vertex form? Answer: y = (x - 2)² - 3 Solution: y = a(x - h)² + k where (h, k) is the vertex. We are told the vertex is (2, -3), so h = 2 and k = -3. y = a(x - 2)² - 3 The parabola passes through (4, 1).
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Recall vertex form of a quadratic** The vertex form of a quadratic function is: y = a(x - h)² + k where (h, k) is the vertex. --- **Step 2: Substitute the vertex into the equation** We are told the vertex is (2, -3), so h = 2 and k = -3. Substitute: y = a(x - 2)² - 3 --- **Step 3: Use the given point to find 'a'** The parabola passes through (4, 1). Substitute x = 4 and y = 1 into the equation: 1 = a(4 - 2)² - 3 --- **Step 4: Simplify and solve for a** 1 = a(2)² - 3 1 = a(4) - 3 1 + 3 = 4a 4 = 4a a = 1 --- **Step 5: Write the final equation** Substitute a = 1, h = 2, k = -3 into vertex form: y = 1(x - 2)² - 3 We usually write it without the 1: y = (x - 2)² - 3 --- **Final answer:** y = (x - 2)² - 3

4. Aisha is designing a rectangular garden with a fixed perimeter of 40 meters. She wants to model the area of the garden as a function of its width. If the width is represented by x meters, what is the maximum possible area the garden can have? Answer: 100 Solution: Let the width be x meters. Since the perimeter is 40 meters, we have: 2(length + width) = 40. Solve for length: length + x = 20, so length = 20 - x.
   Full step-by-step solution

   Step 1: Let the width be x meters. Since the perimeter is 40 meters, we have: 2(length + width) = 40. Step 2: Solve for length: length + x = 20, so length = 20 - x. Step 3: The area A is given by A(x) = length × width = (20 - x) × x = 20x - x². Step 4: This is a quadratic function A(x) = -x² + 20x, which opens downwards. The maximum occurs at the vertex. Step 5: The x-coordinate of the vertex is given by x = -b/(2a) = -20/(2 × -1) = 10. Step 6: Substitute x = 10 into the area function: A(10) = 20(10) - (10)² = 200 - 100 = 100. Step 7: Therefore, the maximum possible area is 100 square meters.

5. f(x) = 2x² - 8x + 6; find f(3) = ? Answer: 0 Solution: f(x) = 2x² - 8x + 6 We need to find f(3). Substitute x = 3 into the function. f(3) = 2*(3)² - 8*(3) + 6 Calculate the exponent first.
   Full step-by-step solution

   We are given the function: f(x) = 2x² - 8x + 6 We need to find f(3). Step 1: Substitute x = 3 into the function. f(3) = 2*(3)² - 8*(3) + 6 Step 2: Calculate the exponent first. (3)² = 9 So f(3) = 2*9 - 8*3 + 6 Step 3: Perform the multiplications. 2*9 = 18 8*3 = 24 So f(3) = 18 - 24 + 6 Step 4: Perform the addition and subtraction from left to right. 18 - 24 = -6 -6 + 6 = 0 Final answer: f(3) = 0

6. f(x) = x² - 4x + 3; f(5) = ? Answer: 8 Solution: Start with the function f(x) = x² - 4x + 3 Substitute x = 5 into the function: f(5) = (5)² - 4(5) + 3 Calculate the exponent: (5)² = 25 Calculate the multiplication: -4(5) = -20 Combine all terms: 25 - 20 + 3 Perform the operations from left to right: 25 - 20 = 5, then 5 + 3 = 8 The answer is 8.
   Full step-by-step solution

   Step 1: Start with the function f(x) = x² - 4x + 3 Step 2: Substitute x = 5 into the function: f(5) = (5)² - 4(5) + 3 Step 3: Calculate the exponent: (5)² = 25 Step 4: Calculate the multiplication: -4(5) = -20 Step 5: Combine all terms: 25 - 20 + 3 Step 6: Perform the operations from left to right: 25 - 20 = 5, then 5 + 3 = 8 The answer is 8.

7. A quadratic function is graphed on a coordinate plane. The parabola opens downward and has x-intercepts at (-2, 0) and (6, 0). The graph passes through the point (2, 16). What is the equation of this quadratic function in factored form? Answer: y = -2(x + 2)(x - 6) Solution: Since the x-intercepts are at (-2, 0) and (6, 0), the factored form is y = a(x + 2)(x - 6). Substitute the point (2, 16) into the equation: 16 = a(2 + 2)(2 - 6). Simplify the expression: 16 = a(4)(-4).
   Full step-by-step solution

   Step 1: Since the x-intercepts are at (-2, 0) and (6, 0), the factored form is y = a(x + 2)(x - 6). Step 2: Substitute the point (2, 16) into the equation: 16 = a(2 + 2)(2 - 6). Step 3: Simplify the expression: 16 = a(4)(-4). Step 4: Calculate: 16 = a(-16). Step 5: Solve for a: a = 16/(-16) = -1. Step 6: Write the final equation: y = -1(x + 2)(x - 6). The answer is y = -(x + 2)(x - 6).

8. Aroha is designing a water fountain for a public park. The water jet follows a parabolic path modeled by the function h(x) = -0.4x² + 8x + 2, where h represents the height of the water in feet above the nozzle and x represents the horizontal distance in feet from the nozzle. The fountain's designers want to install a sensor that only activates when the water is at least 18 feet high. Determine the range of horizontal distances (in feet) from the nozzle where the water reaches or exceeds 18 feet. Answer: 5 ≤ x ≤ 15 Solution: Set the height function equal to 18: -0.4x² + 8x + 2 = 18 Subtract 18 from both sides: -0.4x² + 8x - 16 = 0 Multiply through by -1: 0.4x² - 8x + 16 = 0 Multiply by 5 to eliminate decimal: 2x² - 40x + 80 = 0 Divide by 2: x² - 20x + 40 = 0 Use the quadratic formula: x = [20 ± sqrt(400 - 160)] / 2…
   Full step-by-step solution

   Step 1: Set the height function equal to 18: -0.4x² + 8x + 2 = 18 Step 2: Subtract 18 from both sides: -0.4x² + 8x - 16 = 0 Step 3: Multiply through by -1: 0.4x² - 8x + 16 = 0 Step 4: Multiply by 5 to eliminate decimal: 2x² - 40x + 80 = 0 Step 5: Divide by 2: x² - 20x + 40 = 0 Step 6: Use the quadratic formula: x = [20 ± sqrt(400 - 160)] / 2 = [20 ± sqrt(240)] / 2 = [20 ± 4*sqrt(15)] / 2 = 10 ± 2*sqrt(15) Step 7: Approximate sqrt(15) ≈ 3.873, so x ≈ 10 ± 7.746, giving x ≈ 2.254 and x ≈ 17.746 Step 8: Since the parabola opens downward (a = -0.4 < 0), the water is at least 18 feet high between these two x-values. Step 9: Rounding to the nearest foot, the horizontal distance range is from about 2 feet to about 18 feet. But the problem asks for exact values: the water is ≥ 18 feet when x is between 10 - 2*sqrt(15) and 10 + 2*sqrt(15). In simplified radical form, the answer is 10 - 2*sqrt(15) ≤ x ≤ 10 + 2*sqrt(15). Step 10: Converting to decimal: 10 - 2*3.873 = 10 - 7.746 = 2.254 and 10 + 2*3.873 = 10 + 7.746 = 17.746. So approximately 2.25 ≤ x ≤ 17.75. The exact range is x such that 10 - 2*sqrt(15) ≤ x ≤ 10 + 2*sqrt(15). For practical purposes, the water is at least 18 feet high from approximately 2.25 feet to 17.75 feet horizontally from the nozzle.