Function Concepts

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Answer Key & Explanations

Function Concepts · Grade 8 · Worksheet 2

1. Is the relation {(9, 12), (11, 15), (9, 18), (13, 21)} a function? Answer: No Solution: List the inputs (x-values): 9, 11, 9, 13. Check if any input appears more than once. The input 9 appears twice.
   Full step-by-step solution

   Step 1: List the inputs (x-values): 9, 11, 9, 13. Step 2: Check if any input appears more than once. The input 9 appears twice. Step 3: For input 9, the outputs are 12 and 18. Since one input has two different outputs, this relation does not satisfy the definition of a function (each input must have exactly one output). The answer is No.

2. Isabella is collecting data for her science project on the relationship between the number of hours of sunlight a plant receives each day and the number of flowers it produces. She records the following ordered pairs, where the first number is hours of sunlight and the second number is the number of flowers: (4, 7), (5, 9), (4, 8), (6, 11), (5, 9). Determine whether this set of ordered pairs represents a function. Explain why or why not. Answer: No, it is not a function because the input 4 is paired with two different outputs (7 and 8). Solution: List the ordered pairs: (4, 7), (5, 9), (4, 8), (6, 11), (5, 9). Identify the inputs (first numbers): 4, 5, 4, 6, 5. Check if any input appears more than once.
   Full step-by-step solution

   Step 1: List the ordered pairs: (4, 7), (5, 9), (4, 8), (6, 11), (5, 9). Step 2: Identify the inputs (first numbers): 4, 5, 4, 6, 5. Step 3: Check if any input appears more than once. The input 4 appears twice: once paired with output 7 and once paired with output 8. Step 4: According to the definition of a function, each input must have exactly one output. Since input 4 has two different outputs (7 and 8), the relation is not a function. The answer is: No, it is not a function because the input 4 is paired with two different outputs (7 and 8).

3. A scientist is studying bacterial growth. The number of bacteria after t hours is given by the function f(t) = 2.5 × 10³ × 4^t. How many bacteria were present at the start of the experiment (when t = 0)? Answer: 2500 Solution: f(t) = 2.5 × 10³ × 4^t We need to find the number of bacteria when t = 0, which is the start of the experiment. Substitute t = 0 into the function f(0) = 2.5 × 10³ × 4^0 Any nonzero number raised to the power 0 is 1.
   Full step-by-step solution

   Step 1: Understand the problem The number of bacteria after t hours is given by: f(t) = 2.5 × 10³ × 4^t We need to find the number of bacteria when t = 0, which is the start of the experiment. Step 2: Substitute t = 0 into the function f(0) = 2.5 × 10³ × 4^0 Step 3: Recall exponent rule Any nonzero number raised to the power 0 is 1. So 4^0 = 1. Step 4: Simplify the expression f(0) = 2.5 × 10³ × 1 f(0) = 2.5 × 10³ Step 5: Calculate 2.5 × 10³ 10³ means 1000. So 2.5 × 1000 = 2500. Step 6: Final answer At the start of the experiment (t = 0), there were 2500 bacteria.

4. Emma is designing a triangular banner for a school event. The banner has a base length of 8 feet and a height of 6 feet. She wants to create a second, similar banner where the height is 9 feet. What will be the base length of the second banner? Answer: 12 Solution: The original banner has a base of 8 feet and a height of 6 feet. The ratio of base to height is 8/6, which simplifies to 4/3. The new banner is similar, so it will have the same ratio of base to height.
   Full step-by-step solution

   Step 1: The original banner has a base of 8 feet and a height of 6 feet. The ratio of base to height is 8/6, which simplifies to 4/3. Step 2: The new banner is similar, so it will have the same ratio of base to height. The new height is 9 feet. Step 3: Set up the proportion: base / 9 = 4/3. Step 4: Solve for the base: base = 9 * (4/3). Step 5: Calculate: 9 * 4 = 36, then 36 / 3 = 12. The base length of the second banner is 12 feet.

5. f(x) = 4x² - 5x + 2, find f(3) = ? Answer: 23 Solution: Start with the function f(x) = 4x² - 5x + 2 Substitute x = 3 into the function: f(3) = 4(3)² - 5(3) + 2 Calculate the exponent first: 3² = 9 Multiply: 4 × 9 = 36 and -5 × 3 = -15 Combine all terms: 36 - 15 + 2 Perform the operations from left to right: 36 - 15 = 21, then 21 + 2 = 23 The answer…
   Full step-by-step solution

   Step 1: Start with the function f(x) = 4x² - 5x + 2 Step 2: Substitute x = 3 into the function: f(3) = 4(3)² - 5(3) + 2 Step 3: Calculate the exponent first: 3² = 9 Step 4: Multiply: 4 × 9 = 36 and -5 × 3 = -15 Step 5: Combine all terms: 36 - 15 + 2 Step 6: Perform the operations from left to right: 36 - 15 = 21, then 21 + 2 = 23 The answer is 23.

6. Matiu is tracking the daily high temperatures in his city over a week. He records the temperatures in degrees Celsius for each day: Monday (22), Tuesday (25), Wednesday (22), Thursday (28), Friday (25), Saturday (30), Sunday (25). Matiu wants to know if this set of data represents a function if we consider the day of the week as the input and the temperature as the output. Is this relation a function? Explain why or why not. Answer: Yes, it is a function because each input (day of the week) has exactly one output (temperature). Solution: Identify the inputs and outputs. Inputs are the days of the week (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday). Outputs are the temperatures (22, 25, 22, 28, 25, 30, 25).
   Full step-by-step solution

   Step 1: Identify the inputs and outputs. Inputs are the days of the week (Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday). Outputs are the temperatures (22, 25, 22, 28, 25, 30, 25). Step 2: Check if each input has exactly one output. Monday maps to 22, Tuesday maps to 25, Wednesday maps to 22, Thursday maps to 28, Friday maps to 25, Saturday maps to 30, Sunday maps to 25. Step 3: No day is repeated with a different temperature. Each day has only one temperature recorded. Step 4: Therefore, this relation satisfies the definition of a function: each input has exactly one output. The answer is: Yes, it is a function.

7. f(x) = 6x² - 11x + 1, find f(6) = ? Answer: 151 Solution: Write the function: f(x) = 6x² - 11x + 1 Substitute x = 6: f(6) = 6(6)² - 11(6) + 1 Calculate the exponent: (6)² = 36, so f(6) = 6(36) - 11(6) + 1 Perform the multiplications: 6 × 36 = 216 and 11 × 6 = 66, so f(6) = 216 - 66 + 1 Subtract: 216 - 66 = 150 Add: 150 + 1 = 151 The answer is 151.
   Full step-by-step solution

   Step 1: Write the function: f(x) = 6x² - 11x + 1 Step 2: Substitute x = 6: f(6) = 6(6)² - 11(6) + 1 Step 3: Calculate the exponent: (6)² = 36, so f(6) = 6(36) - 11(6) + 1 Step 4: Perform the multiplications: 6 × 36 = 216 and 11 × 6 = 66, so f(6) = 216 - 66 + 1 Step 5: Subtract: 216 - 66 = 150 Step 6: Add: 150 + 1 = 151 The answer is 151.

8. Is the relation {(1, 6), (6, 1), (1, 11), (11, 1)} a function? Explain why or why not. Answer: No, it is not a function because the input 1 has two different outputs: 6 and 11. Solution: List all ordered pairs: (1, 6), (6, 1), (1, 11), (11, 1). Identify the input values (first numbers): 1, 6, 1, 11. Check if any input appears more than once.
   Full step-by-step solution

   Step 1: List all ordered pairs: (1, 6), (6, 1), (1, 11), (11, 1). Step 2: Identify the input values (first numbers): 1, 6, 1, 11. Step 3: Check if any input appears more than once. The input 1 appears twice. Step 4: Look at the outputs for input 1: the first pair gives output 6, the third pair gives output 11. Step 5: Since input 1 has two different outputs (6 and 11), the relation does not satisfy the definition of a function. The answer is: No, it is not a function because the input 1 has two different outputs: 6 and 11.