Function Concepts

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Answer Key & Explanations

Function Concepts · Grade 8 · Worksheet 1

1. Is the relation {(3, 7), (5, 9), (3, 11), (7, 13)} a function? Answer: No Solution: List the inputs: 3, 5, 3, 7. Notice that the input 3 appears twice: once with output 7 and once with output 11. A function requires each input to have exactly one output.
   Full step-by-step solution

   Step 1: List the inputs: 3, 5, 3, 7. Step 2: Notice that the input 3 appears twice: once with output 7 and once with output 11. Step 3: A function requires each input to have exactly one output. Since input 3 has two different outputs, this relation is not a function. The answer is No.

2. Noah is collecting data on how many hours each of his classmates spends on homework per week. He records the following ordered pairs, where the first number is the student's ID and the second number is the hours spent: {(101, 6), (102, 11), (103, 6), (104, 16), (105, 11)}. Does this relation represent a function? Explain why or why not. Answer: Yes, it is a function because each student ID (input) has exactly one number of hours (output). Solution: Identify the inputs (student IDs) and outputs (hours). The inputs are: 101, 102, 103, 104, 105. The outputs are: 6, 11, 6, 16, 11.
   Full step-by-step solution

   Step 1: Identify the inputs (student IDs) and outputs (hours). The inputs are: 101, 102, 103, 104, 105. The outputs are: 6, 11, 6, 16, 11. Step 2: Check if any input repeats. Each input appears only once: 101 appears once, 102 once, 103 once, 104 once, 105 once. Step 3: Since each input has exactly one output, the relation is a function. The answer is: Yes, it is a function because each student ID (input) has exactly one number of hours (output).

3. Aroha draws a mapping diagram on her whiteboard. The domain (inputs) contains the numbers 3, 5, 7, and 9. The range (outputs) contains the numbers 9, 15, 21, 27, and 33. She draws arrows as follows: - From 3 to 9 - From 5 to 15 - From 7 to 21 - From 9 to 27 - From 9 to 33 Is this relation a function? Explain why or why not. Answer: No, it is not a function. Solution: Recall the definition of a function: a relation is a function if and only if each input has exactly one output. List the inputs (domain): 3, 5, 7, 9.
   Full step-by-step solution

   Step 1: Recall the definition of a function: a relation is a function if and only if each input has exactly one output. Step 2: List the inputs (domain): 3, 5, 7, 9. Step 3: Examine the arrows from each input: - Input 3 maps to output 9 (one arrow) - Input 5 maps to output 15 (one arrow) - Input 7 maps to output 21 (one arrow) - Input 9 maps to output 27 and also to output 33 (two arrows) Step 4: Since input 9 has two different outputs (27 and 33), this violates the function rule that each input must have exactly one output. Step 5: Therefore, the relation is NOT a function. The answer is: No, it is not a function.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A second triangle is created by rotating the original triangle 90 degrees counterclockwise about the origin. What are the coordinates of the vertex that was originally at (6,8) after this rotation? Answer: (-8,6) Solution: A = (0, 0) B = (6, 0) C = (6, 8) We are rotating the triangle 90° counterclockwise about the origin (0, 0). We need the new coordinates of the vertex that was originally at (6, 8) after rotation.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the problem** We have a right triangle with vertices: A = (0, 0) B = (6, 0) C = (6, 8) We are rotating the triangle 90° counterclockwise about the origin (0, 0). We need the new coordinates of the vertex that was originally at (6, 8) after rotation. --- **Step 2: Recall the rotation rule** For a point (x, y) rotated 90° counterclockwise about the origin: The new coordinates are: x' = -y y' = x That is: (x, y) → (-y, x) --- **Step 3: Apply the rule to point (6, 8)** Original point: (6, 8) x = 6, y = 8 After rotation: x' = -y = -8 y' = x = 6 So new coordinates: (-8, 6) --- **Step 4: Verify with reasoning** - The original point (6, 8) is in the first quadrant (right and up from origin). - After 90° counterclockwise rotation, it should end up in the second quadrant (left and up from origin). - The distance from the origin should be preserved: Original distance = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 New distance = sqrt((-8)^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 ✓ --- **Final answer:** (-8, 6)

5. A scientist is studying the growth of a special type of crystal. The volume of the crystal after t days is given by the function V(t) = 8.2 × 10⁻⁶ × 2^t cubic centimeters. What is the volume of the crystal after 5 days? Express your answer in scientific notation. Answer: 2.624×10⁻⁴ Solution: The function is V(t) = 8.2 × 10⁻⁶ × 2^t Substitute t = 5 into the function: V(5) = 8.2 × 10⁻⁶ × 2^5 Calculate 2^5 = 32 Multiply 8.2 × 32 = 262.4 Now we have 262.4 × 10⁻⁶ Convert to proper scientific notation: 262.4 × 10⁻⁶ = 2.624 × 10² × 10⁻⁶ Combine exponents: 2.624 × 10^(2-6) = 2.624 × 10⁻⁴…
   Full step-by-step solution

   Step 1: The function is V(t) = 8.2 × 10⁻⁶ × 2^t Step 2: Substitute t = 5 into the function: V(5) = 8.2 × 10⁻⁶ × 2^5 Step 3: Calculate 2^5 = 32 Step 4: Multiply 8.2 × 32 = 262.4 Step 5: Now we have 262.4 × 10⁻⁶ Step 6: Convert to proper scientific notation: 262.4 × 10⁻⁶ = 2.624 × 10² × 10⁻⁶ Step 7: Combine exponents: 2.624 × 10^(2-6) = 2.624 × 10⁻⁴ The answer is 2.624×10⁻⁴.

6. f(x) = 3x² - 2x + 1, find f(-2) = ? Answer: 17 Solution: Write the function: f(x) = 3x² - 2x + 1 Substitute x = -2: f(-2) = 3(-2)² - 2(-2) + 1 Calculate the exponent first: (-2)² = 4 Multiply: 3 × 4 = 12 and -2 × (-2) = 4 Rewrite the expression: 12 + 4 + 1 Add from left to right: 12 + 4 = 16, then 16 + 1 = 17 The final answer is 17.
   Full step-by-step solution

   Step 1: Write the function: f(x) = 3x² - 2x + 1 Step 2: Substitute x = -2: f(-2) = 3(-2)² - 2(-2) + 1 Step 3: Calculate the exponent first: (-2)² = 4 Step 4: Multiply: 3 × 4 = 12 and -2 × (-2) = 4 Step 5: Rewrite the expression: 12 + 4 + 1 Step 6: Add from left to right: 12 + 4 = 16, then 16 + 1 = 17 Step 7: The final answer is 17.

7. Aisha is comparing two internet service providers. Provider A charges a $25 monthly fee plus $0.50 per gigabyte of data used. Provider B charges a $40 monthly fee plus $0.25 per gigabyte of data used. After how many gigabytes of data usage will the total monthly cost be the same for both providers? Answer: 60 Solution: Let x represent the number of gigabytes used. Write the cost equation for Provider A: Cost_A = 25 + 0.50x Write the cost equation for Provider B: Cost_B = 40 + 0.25x Set the costs equal to find when they are the same: 25 + 0.50x = 40 + 0.25x Subtract 0.25x from both sides: 25 + 0.25x = 40…
   Full step-by-step solution

   Step 1: Let x represent the number of gigabytes used. Step 2: Write the cost equation for Provider A: Cost_A = 25 + 0.50x Step 3: Write the cost equation for Provider B: Cost_B = 40 + 0.25x Step 4: Set the costs equal to find when they are the same: 25 + 0.50x = 40 + 0.25x Step 5: Subtract 0.25x from both sides: 25 + 0.25x = 40 Step 6: Subtract 25 from both sides: 0.25x = 15 Step 7: Divide both sides by 0.25: x = 15 / 0.25 Step 8: Calculate: x = 60 The answer is 60 gigabytes.

8. Is the relation {(8, 15), (12, 20), (8, 25), (10, 18)} a function? Explain why or why not. Answer: No, because the input 8 has two different outputs: 15 and 25. Solution: List the input-output pairs: (8,15), (12,20), (8,25), (10,18). Identify each input: 8, 12, 8, 10. Notice that the input 8 appears twice: once with output 15 and once with output 25.
   Full step-by-step solution

   Step 1: List the input-output pairs: (8,15), (12,20), (8,25), (10,18). Step 2: Identify each input: 8, 12, 8, 10. Step 3: Notice that the input 8 appears twice: once with output 15 and once with output 25. Step 4: For a relation to be a function, each input must have exactly one output. Since input 8 has two different outputs, this relation is not a function. The answer is: No, because the input 8 has two different outputs: 15 and 25.