Domain and Range

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Domain and Range · Grade 9 · Worksheet 1

1. Matiu is designing a water fountain for a park. The height of the water stream above the fountain basin is modeled by the function h(x) = -0.25x² + 4x, where h is the height in feet and x is the horizontal distance in feet from the nozzle. The fountain's basin extends 16 feet horizontally from the nozzle. The park requires that the water must be at least 6 feet high for visual effect. Graph the function and determine the domain and range of h(x) for this fountain design, considering the basin's edge and the requirement that height cannot be negative. Answer: Domain: 0 ≤ x ≤ 16, Range: 0 ≤ h(x) ≤ 16 Solution: The function is h(x) = -0.25x² + 4x. This is a quadratic that opens downward (since coefficient of x² is negative).
   Full step-by-step solution

   Step 1: The function is h(x) = -0.25x² + 4x. This is a quadratic that opens downward (since coefficient of x² is negative). Step 2: Find the x-intercepts (where h = 0): Set -0.25x² + 4x = 0 Factor out x: x(-0.25x + 4) = 0 So x = 0 or -0.25x + 4 = 0 → x = 16 The water stream starts at x = 0 (nozzle) and hits the basin at x = 16 feet. Step 3: Domain: The horizontal distance from the nozzle ranges from 0 to 16 feet, so domain is 0 ≤ x ≤ 16. Step 4: Find the vertex (maximum height). The vertex x-coordinate is at x = -b/(2a) where a = -0.25 and b = 4. x = -4/(2*(-0.25)) = -4/(-0.5) = 8 Step 5: Find the maximum height: h(8) = -0.25*(8)² + 4*8 = -0.25*64 + 32 = -16 + 32 = 16 feet. Step 6: The height ranges from 0 feet (at x = 0 and x = 16) up to 16 feet (at x = 8), so range is 0 ≤ h(x) ≤ 16. The domain is 0 ≤ x ≤ 16 and the range is 0 ≤ h(x) ≤ 16.

2. f(x) = 4x² - 7x + 5; f(2) = ? Answer: 7 Solution: Start with the function f(x) = 4x² - 7x + 5 Substitute x = 2 into the function: f(2) = 4(2)² - 7(2) + 5 Calculate the exponent first: (2)² = 4 Multiply: 4 × 4 = 16 and -7 × 2 = -14 Combine all terms: 16 - 14 + 5 Perform the operations from left to right: 16 - 14 = 2, then 2 + 5 = 7 The answer is 7.
   Full step-by-step solution

   Step 1: Start with the function f(x) = 4x² - 7x + 5 Step 2: Substitute x = 2 into the function: f(2) = 4(2)² - 7(2) + 5 Step 3: Calculate the exponent first: (2)² = 4 Step 4: Multiply: 4 × 4 = 16 and -7 × 2 = -14 Step 5: Combine all terms: 16 - 14 + 5 Step 6: Perform the operations from left to right: 16 - 14 = 2, then 2 + 5 = 7 The answer is 7.

3. Maya is designing a skateboard ramp with a parabolic shape. The height of the ramp above the ground is modeled by the function h(x) = -0.5x² + 4x, where h is the height in feet and x is the horizontal distance in feet from the starting point. What is the maximum height the ramp reaches, and at what horizontal distance does this occur? Answer: 8 feet at 4 feet Solution: The function is h(x) = -0.5x² + 4x. Since the coefficient of x² is negative (-0.5), the parabola opens downward, so the vertex gives the maximum height.
   Full step-by-step solution

   Step 1: The function is h(x) = -0.5x² + 4x. Since the coefficient of x² is negative (-0.5), the parabola opens downward, so the vertex gives the maximum height. Step 2: To find the x-coordinate of the vertex, use the formula x = -b/(2a), where a = -0.5 and b = 4. Step 3: Calculate x = -4/(2 × -0.5) = -4/(-1) = 4 feet. Step 4: Substitute x = 4 into the function to find the maximum height: h(4) = -0.5(4)² + 4(4) = -0.5(16) + 16 = -8 + 16 = 8 feet. Step 5: Therefore, the maximum height is 8 feet, and it occurs at a horizontal distance of 4 feet from the starting point.

4. Emma is modeling the height of a water fountain's spray. The height h(x) in meters of the water at a horizontal distance x meters from the nozzle is given by the quadratic function h(x) = -3x² + 18x + 5. The fountain's spray forms a continuous parabolic arc above ground. A bird is flying at a constant height of 15 meters above the ground and will only be visible in the fountain's spray when the water reaches at least that height. Determine the domain of x-values for which the water's height is at least 15 meters, and state the corresponding range of water heights over that interval. Answer: Domain: 2/3 ≤ x ≤ 16/3; Range: 15 ≤ h(x) ≤ 32 Solution: Set h(x) = 15 to find where the water reaches 15 meters: -3x² + 18x + 5 = 15. Step 2: Rearrange to -3x² + 18x - 10 = 0. Multiply by -1: 3x² - 18x + 10 = 0.
   Full step-by-step solution

   Step 1: Set h(x) = 15 to find where the water reaches 15 meters: -3x² + 18x + 5 = 15. Step 2: Rearrange to -3x² + 18x - 10 = 0. Multiply by -1: 3x² - 18x + 10 = 0. Step 3: Use quadratic formula: x = [18 ± sqrt(324 - 120)] / (2*3) = [18 ± sqrt(204)] / 6 = [18 ± 2*sqrt(51)] / 6 = [9 ± sqrt(51)] / 3. Step 4: Since sqrt(51) is between 7 and 8, approximate: sqrt(51) ≈ 7.141, so x ≈ (9 - 7.141)/3 = 1.859/3 ≈ 0.62 and x ≈ (9 + 7.141)/3 = 16.141/3 ≈ 5.38. Step 5: As exact values: x = (9 - sqrt(51))/3 and x = (9 + sqrt(51))/3. The domain where h(x) ≥ 15 is (9 - sqrt(51))/3 ≤ x ≤ (9 + sqrt(51))/3. Step 6: Find the vertex: x_vertex = -18/(2*(-3)) = 3. h(3) = -3(9) + 18(3) + 5 = -27 + 54 + 5 = 32. Step 7: Since the parabola opens downward, the maximum height is 32 m. Over the domain interval, the range is from 15 up to 32. So range: 15 ≤ h(x) ≤ 32. Step 8: The exact domain in simplified form: (9 - sqrt(51))/3 ≤ x ≤ (9 + sqrt(51))/3. Note: (9 - sqrt(51))/3 = 3 - sqrt(51)/3 and (9 + sqrt(51))/3 = 3 + sqrt(51)/3. For a cleaner exact form: Domain: [3 - sqrt(51)/3, 3 + sqrt(51)/3]. Range: [15, 32].

5. f(x) = -2x² + 12x - 10; f(4) = ? Answer: 6 Solution: Substitute x = 4 into the function: f(4) = -2(4)² + 12(4) - 10 Calculate the exponent first: (4)² = 16 Multiply: -2 × 16 = -32 and 12 × 4 = 48 Rewrite the expression: f(4) = -32 + 48 - 10 Add from left to right: -32 + 48 = 16 Subtract: 16 - 10 = 6 The answer is 6.
   Full step-by-step solution

   Step 1: Substitute x = 4 into the function: f(4) = -2(4)² + 12(4) - 10 Step 2: Calculate the exponent first: (4)² = 16 Step 3: Multiply: -2 × 16 = -32 and 12 × 4 = 48 Step 4: Rewrite the expression: f(4) = -32 + 48 - 10 Step 5: Add from left to right: -32 + 48 = 16 Step 6: Subtract: 16 - 10 = 6 The answer is 6.

6. f(x) = 3x² - 5x + 7; f(-1) = ? Answer: 15 Solution: Start with the function f(x) = 3x² - 5x + 7 Substitute x = -1 into the function: f(-1) = 3(-1)² - 5(-1) + 7 Calculate the exponent first: (-1)² = 1 Multiply: 3 × 1 = 3 Multiply: -5 × (-1) = 5 Now we have: 3 + 5 + 7 Add from left to right: 3 + 5 = 8, then 8 + 7 = 15 The final answer is 15.
   Full step-by-step solution

   Step 1: Start with the function f(x) = 3x² - 5x + 7 Step 2: Substitute x = -1 into the function: f(-1) = 3(-1)² - 5(-1) + 7 Step 3: Calculate the exponent first: (-1)² = 1 Step 4: Multiply: 3 × 1 = 3 Step 5: Multiply: -5 × (-1) = 5 Step 6: Now we have: 3 + 5 + 7 Step 7: Add from left to right: 3 + 5 = 8, then 8 + 7 = 15 Step 8: The final answer is 15.

7. f(x) = 4x² - 7x + 2; f(2) = ? Answer: 4 Solution: Start with the function f(x) = 4x² - 7x + 2 Substitute x = 2 into the function: f(2) = 4(2)² - 7(2) + 2 Calculate the exponent: (2)² = 4 Multiply: 4 × 4 = 16 and -7 × 2 = -14 Combine all terms: 16 - 14 + 2 Simplify: 16 - 14 = 2, then 2 + 2 = 4 The answer is 4.
   Full step-by-step solution

   Step 1: Start with the function f(x) = 4x² - 7x + 2 Step 2: Substitute x = 2 into the function: f(2) = 4(2)² - 7(2) + 2 Step 3: Calculate the exponent: (2)² = 4 Step 4: Multiply: 4 × 4 = 16 and -7 × 2 = -14 Step 5: Combine all terms: 16 - 14 + 2 Step 6: Simplify: 16 - 14 = 2, then 2 + 2 = 4 The answer is 4.