Functional Relationships

Grade 8 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

Functional Relationships: The Input-Output Machine 🧮

A functional relationship shows how one quantity (the input) determines another quantity (the output). Think of it like a vending machine: you select a code (input), and it gives you a specific snack (output). We write this as y = f(x), where 'x' is the input and 'y' is the output.

How to Work with Functions

  1. Identify the Rule: Look at the equation (e.g., y = 2x + 1). This tells you what to do to the input (x).
  2. Choose an Input: Pick a value for x. It's often helpful to make a table.
  3. Calculate the Output: Substitute your input into the equation and solve for y.
  4. Plot the Points: Use your (x, y) pairs to create a graph. The graph shows all possible input-output combinations!

Worked Examples

Example 1: Linear Function
Rule: y = 3x - 2
If x = 4, then y = 3(4) - 2 = 12 - 2 = 10.
So, the ordered pair is (4, 10).

Example 2: Creating a Table
For y = x + 5:
If x = 0, y = 5 → (0, 5)
If x = 1, y = 6 → (1, 6)
If x = 2, y = 7 → (2, 7)
Plot these points and draw a line through them.

Common Mistakes to Avoid ⚠️

  • Mixing up x and y: Remember, x is always the input (independent), y is the output (dependent).
  • Order of Operations: In y = 2x + 1, you must multiply x by 2 before adding 1.
  • Graphing Errors: The point (3, 2) means go RIGHT 3, then UP 2. Order matters!

Tips & Tricks

  • Machine Mindset: Imagine the function as a machine that changes the input.
  • Pattern Spotting: In a table, see how y changes when x changes. If it's constant, it's linear!
  • Vertical Line Test: If a vertical line hits a graph in more than one place, it's NOT a function.

How to Practice

  • Create input-output tables for different rules like y = 4x and y = x - 3.
  • Use online graphing tools to see how changing the rule changes the graph.
  • Find real-world examples: The cost of pizza is a function of the number of toppings!

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 3² × (4 + 2) - 5 = ?

Hint: Remember to follow the order of operations - start with calculations inside parentheses, then handle exponents, then multiplication and division from left to right, and finally addition and subtraction.

Show the answer

Answer: 49

  1. Identify the expression 3² × (4 + 2) - 5
  2. Handle parentheses first Inside parentheses: 4 + 2 = 6 Now we have: 3² × 6 - 5
  3. Evaluate exponents 3² means 3 × 3 = 9 Now we have: 9 × 6 - 5
  4. Perform multiplication 9 × 6 = 54 Now we have: 54 - 5
  5. Perform subtraction 54 - 5 = 49 Final answer: 49

Let's solve this step by step using the order of operations (PEMDAS/BODMAS).

2 2³ × (4² - 3²) ÷ 5 = ?

Hint: Remember to follow the order of operations: parentheses first, then exponents, then multiplication and division from left to right.

Show the answer

Answer: 11.2

  1. Calculate the exponents** 2³ = 2 × 2 × 2 = 8 4² = 4 × 4 = 16 3² = 3 × 3 = 9 So the expression becomes: 8 × (16 - 9) ÷ 5 --- **
  2. Simplify inside the parentheses** 16 - 9 = 7 Now we have: 8 × 7 ÷ 5 --- **
  3. Multiply** 8 × 7 = 56 Now we have: 56 ÷ 5 --- **
  4. Divide** 56 ÷ 5 = 11.2 --- **Final Answer:** 11.2

Let's solve step by step. We have: 2³ × (4² - 3²) ÷ 5 --- **

3 (2.4 × 10⁵) ÷ (6 × 10²) = ?

Hint: Separate the coefficients and exponents, then divide each part separately before combining the results.

Show the answer

Answer: 400

  1. Write the expression as (2.4 ÷ 6) × (10⁵ ÷ 10²)
  2. Divide the coefficients: 2.4 ÷ 6 = 0.4
  3. Divide the powers of 10: 10⁵ ÷ 10² = 10^(5-2) = 10³
  4. Multiply the results: 0.4 × 10³ = 0.4 × 1000 = 400

The answer is 400.

4 (2.5 × 10⁸) ÷ (5 × 10³) = ?

Hint: Consider separating the coefficients and exponents when dividing numbers in scientific notation

Show the answer

Answer: 50000

  1. Separate the coefficients and exponents: (2.5 ÷ 5) × (10⁸ ÷ 10³)
  2. Divide the coefficients: 2.5 ÷ 5 = 0.5
  3. Divide the powers of 10: 10⁸ ÷ 10³ = 10⁵
  4. Multiply the results: 0.5 × 10⁵ = 5 × 10⁴
  5. Convert to standard form: 5 × 10⁴ = 50,000

The answer is 50000.

5 (3x² - 7x + 2) ÷ (x - 2) = ?

Hint: Use polynomial long division to divide the quadratic expression by the linear binomial. Focus on dividing the leading terms first.

Show the answer

Answer: 3x - 1

  1. Set up the division. We are dividing 3x² - 7x + 2 by x - 2.
  2. Divide the first term of the dividend by the first term of the divisor. Take 3x² and divide by x. 3x² / x = 3x. Write 3x above the division bar.
  3. Multiply the divisor by this result. Multiply (x - 2) by 3x: (x)(3x) = 3x², and (-2)(3x) = -6x. So we get 3x² - 6x.
  4. Subtract this result from the dividend. Write: (3x² - 7x + 2) - (3x² - 6x) = 3x² - 7x + 2 - 3x² + 6x. Combine like terms: 3x² - 3x² = 0, and -7x + 6x = -1x. Bring down the +2: we now have -1x + 2.
  5. Repeat the process with the new polynomial (-1x + 2). Divide the first term of this polynomial by the first term of the divisor: -1x / x = -1. Write -1 above the division bar next to 3x.
  6. Multiply the divisor by this new result. Multiply (x - 2) by -1: (x)(-1) = -1x, and (-2)(-1) = +2. So we get -1x + 2.
  7. Subtract this from the current polynomial. (-1x + 2) - (-1x + 2) = -1x + 2 + 1x - 2 = 0. The remainder is 0. Final result: The quotient is 3x - 1. Answer: 3x - 1

Let's divide (3x² - 7x + 2) by (x - 2) using polynomial long division.

6 (2.4 × 10³) ÷ (6 × 10⁻²) = ?

Hint: When dividing numbers in scientific notation, divide the coefficients and subtract the exponents of 10

Show the answer

Answer: 40000

  1. Divide the coefficients: 2.4 ÷ 6 = 0.4
  2. Subtract the exponents: 3 - (-2) = 3 + 2 = 5
  3. Combine the results: 0.4 × 10⁵
  4. Convert to standard form: 0.4 × 100,000 = 40,000

The answer is 40000.

Practise this topic — 10 free problems, no signup →