Function Concepts

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Answer Key & Explanations

Function Concepts · Grade 8 · Worksheet 3

1. Is the relation {(0, 5), (5, 10), (10, 15), (5, 20)} a function? Answer: No Solution: List the inputs (x-values): 0, 5, 10, 5. Notice that the input 5 appears twice: once with output 10 and once with output 20. A function requires each input to have exactly one output.
   Full step-by-step solution

   Step 1: List the inputs (x-values): 0, 5, 10, 5. Step 2: Notice that the input 5 appears twice: once with output 10 and once with output 20. Step 3: A function requires each input to have exactly one output. Since input 5 has two different outputs, this relation is not a function. The answer is No.

2. Emma is tracking the growth of her sunflower plant. She notices that the height increases by 2.5 inches each week. After 4 weeks, the plant is 18 inches tall. Write a linear function in the form h(w) = mw + b that represents the height of the sunflower after w weeks, where h is the height in inches. Answer: h(w) = 2.5w + 8 Solution: Identify the rate of change (slope). The height increases by 2.5 inches per week, so m = 2.5. Use the given point to find the initial height.
   Full step-by-step solution

   Step 1: Identify the rate of change (slope). The height increases by 2.5 inches per week, so m = 2.5. Step 2: Use the given point to find the initial height. After 4 weeks (w = 4), the height is 18 inches (h = 18). Step 3: Substitute into the function form h(w) = mw + b: 18 = 2.5(4) + b Step 4: Calculate: 18 = 10 + b Step 5: Solve for b: b = 18 - 10 = 8 Step 6: Write the final function: h(w) = 2.5w + 8

3. Olivia is organizing a school science fair. She records the number of projects submitted by each grade level. She notices that the number of projects, p, is related to the grade level, g, by the rule p = 3g - 1. She then lists the following ordered pairs to represent this relation: (1, 2), (2, 5), (3, 8), (4, 11), (5, 14). Is this relation a function? Explain why or why not. Answer: Yes, it is a function because each input (grade level) has exactly one output (number of projects). Solution: Recall the definition of a function: a relation where each input (x-value) has exactly one output (y-value). List the inputs (grade levels) from the ordered pairs: 1, 2, 3, 4, 5.
   Full step-by-step solution

   Step 1: Recall the definition of a function: a relation where each input (x-value) has exactly one output (y-value). Step 2: List the inputs (grade levels) from the ordered pairs: 1, 2, 3, 4, 5. Step 3: Check if any input appears more than once: each input appears exactly once in the list. Step 4: Check if each input has only one output: for input 1, output is 2; for input 2, output is 5; for input 3, output is 8; for input 4, output is 11; for input 5, output is 14. No input has more than one output. Step 5: Also verify with the rule p = 3g - 1: for each g, the formula gives a single value of p. Step 6: Therefore, the relation is a function because every input has exactly one output. The answer is: Yes, it is a function because each input (grade level) has exactly one output (number of projects).

4. Liam is organizing his baseball card collection. He creates a rule that pairs each player's jersey number with the number of home runs that player hit last season. Liam records the following pairs: (7, 15), (15, 22), (21, 15), (7, 9), (33, 41). Based on Liam's rule, is this relation a function? Explain why or why not. Answer: No, it is not a function because the input 7 is paired with two different outputs (15 and 9). Solution: A function requires that each input (the first number in each pair) has exactly one output (the second number). List the inputs: 7, 15, 21, 7, 33 Notice that the input 7 appears twice: once as (7, 15) and once as (7, 9) The input 7 has two different outputs: 15 and 9 Since one input maps to more…
   Full step-by-step solution

   Step 1: A function requires that each input (the first number in each pair) has exactly one output (the second number). Step 2: List the inputs: 7, 15, 21, 7, 33 Step 3: Notice that the input 7 appears twice: once as (7, 15) and once as (7, 9) Step 4: The input 7 has two different outputs: 15 and 9 Step 5: Since one input maps to more than one output, this violates the definition of a function. Step 6: Therefore, this relation is not a function. The answer is: No, it is not a function because the input 7 is paired with two different outputs (15 and 9).

5. Aisha is designing a rectangular banner for a school event. The length of the banner is 5 feet more than its width. If the area of the banner is 84 square feet, what is the width of the banner in feet? Answer: 7 Solution: Let the width be w feet. Then the length is w + 5 feet. The area of a rectangle is length × width, so (w + 5) × w = 84.
   Full step-by-step solution

   Step 1: Let the width be w feet. Then the length is w + 5 feet. Step 2: The area of a rectangle is length × width, so (w + 5) × w = 84. Step 3: Expand the equation: w^2 + 5w = 84. Step 4: Subtract 84 from both sides: w^2 + 5w - 84 = 0. Step 5: Factor the quadratic equation: (w + 12)(w - 7) = 0. Step 6: Solve for w: w = -12 or w = 7. Step 7: Since width cannot be negative, the width is 7 feet. The answer is 7.

6. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. The area of the garden is 65 square meters. What is the width of Liam's garden? Answer: 5 Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: Length \( l = 2w + 3 \). Area of a rectangle = length × width.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define the variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: Length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area of a rectangle = length × width. Given area = 65 m²: \[ (2w + 3) \times w = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** We can use factoring: We need two numbers whose product is \( 2 \times (-65) = -130 \) and whose sum is \( 3 \). Those numbers are \( 13 \) and \( -10 \). So rewrite \( 3w \) as \( 13w - 10w \): \[ 2w^2 + 13w - 10w - 65 = 0 \] Group terms: \[ w(2w + 13) - 5(2w + 13) = 0 \] \[ (2w + 13)(w - 5) = 0 \] --- **Step 5: Find possible values of \( w \)** \[ 2w + 13 = 0 \quad \text{or} \quad w - 5 = 0 \] \[ w = -\frac{13}{2} \quad \text{or} \quad w = 5 \] --- **Step 6: Interpret the solution** Width cannot be negative, so \( w = 5 \) meters. --- **Step 7: Check** Width = 5 m, length = \( 2(5) + 3 = 13 \) m. Area = \( 13 \times 5 = 65 \) m². Correct. --- **Final answer:** The width is 5 meters.

7. Sophia is collecting data on the number of books her classmates read last month. She records the following pairs of data, where the first number is the student ID and the second number is the number of books read: (21, 6), (16, 11), (21, 6), (31, 16), (26, 11). Sophia wonders if this relation represents a function. Based on the definition of a function (each input has exactly one output), is this relation a function? Explain why or why not. Answer: Yes, it is a function. Solution: Identify the inputs (student IDs) and outputs (books read). The pairs are (21, 6), (16, 11), (21, 6), (31, 16), (26, 11). Check if any input repeats.
   Full step-by-step solution

   Step 1: Identify the inputs (student IDs) and outputs (books read). The pairs are (21, 6), (16, 11), (21, 6), (31, 16), (26, 11). Step 2: Check if any input repeats. The student ID 21 appears twice: once with output 6 and again with output 6. Step 3: Since the same input (21) always maps to the same output (6), and all other inputs (16, 31, 26) appear only once, each input has exactly one output. Step 4: Therefore, the relation satisfies the definition of a function. The answer is: Yes, it is a function.