Functions in Context

Grade 9 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

Functions in Context: Making Math Real 🌍

A function is like a math machine: you put a number in (the input), it follows a rule, and gives you a number out (the output). We use functions in context to model and solve real-world situations, like calculating phone bills, predicting profits, or figuring out distances.

How to Solve Function-in-Context Problems

  1. Identify the Variables: Figure out what the input (usually x) and output (usually f(x) or y) represent in the story.
  2. Find the Function Rule: Look for the pattern or equation that connects the variables.
  3. Substitute and Calculate: Plug the given input value into the function to find the output.
  4. Interpret Your Answer: Go back to the story! What does your final number actually mean?

Worked Examples

Example 1: The Taxi Fare

A taxi charges a $3 flat fee plus $2 per mile. The function for the total cost is C(m) = 3 + 2m, where m is the number of miles. How much is a 5-mile ride?

  1. Substitute: C(5) = 3 + 2(5)
  2. Calculate: C(5) = 3 + 10 = 13
  3. Interpret: A 5-mile taxi ride costs $13.

Example 2: The Swimming Pool

A pool is being filled with water. The function V(t) = 50 + 10t gives the volume (in gallons) after t minutes. What is the volume after 6 minutes?

  1. Substitute: V(6) = 50 + 10(6)
  2. Calculate: V(6) = 50 + 60 = 110
  3. Interpret: After 6 minutes, the pool contains 110 gallons of water.

🚨 Common Mistakes to Avoid

  • Forgetting the Units: Always write the units (like $, gallons, miles) in your final answer.
  • Mixing up Input & Output: Remember, f(x) is the result or output when you use x as the input.
  • Ignoring the Context: An answer of "13" is meaningless. You must say "13 dollars."

💡 Tips & Tricks

  • Word Swap: Read "f(5)" as "the output when the input is 5."
  • Check for Reasonableness: Does your answer make sense? A 100-mile taxi ride shouldn't cost $5!
  • Underline Key Info: Circle the numbers and units in the word problem before you start.

How to Practice

To get better, try these activities:

  • Create your own real-world function stories for a friend to solve.
  • Look for functions in your daily life (like a pizza cost: $15 + $2 per topping).
  • Practice with worksheets that mix different contexts to build flexibility.

Practice problems

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

1 f(x) = 3x² - 5x + 2, find f(4) = ?

Hint: Substitute the given value into the function and follow the order of operations carefully.

Show the answer

Answer: 30

  1. Write the function: f(x) = 3x² - 5x + 2
  2. Substitute x = 4: f(4) = 3(4)² - 5(4) + 2
  3. Calculate the exponent: (4)² = 16
  4. Multiply: 3 × 16 = 48
  5. Multiply: 5 × 4 = 20
  6. Substitute back: f(4) = 48 - 20 + 2
  7. Perform subtraction: 48 - 20 = 28
  8. Perform addition: 28 + 2 = 30

The answer is 30.

2 f(x) = 2x³ - 7x + 5, find f(3) = ?

Hint: Substitute the given value into the function and follow the order of operations carefully, remembering to handle exponents before multiplication and subtraction.

Show the answer

Answer: 38

  1. Start with the function f(x) = 2x³ - 7x + 5
  2. Substitute x = 3 into the function: f(3) = 2(3)³ - 7(3) + 5
  3. Calculate the exponent first: (3)³ = 27
  4. Multiply: 2 × 27 = 54 and 7 × 3 = 21
  5. Substitute back: f(3) = 54 - 21 + 5
  6. Perform subtraction and addition from left to right: 54 - 21 = 33, then 33 + 5 = 38

The answer is 38.

3 f(x) = 2x^2 - 8x + 6, find f(3) = ?

Hint: To evaluate a function at a specific value, substitute that value for the variable in the function's expression and simplify using order of operations.

Show the answer

Answer: 0

  1. Start with the function f(x) = 2x^2 - 8x + 6
  2. Substitute x = 3 into the function: f(3) = 2(3)^2 - 8(3) + 6
  3. Calculate the exponent first: 3^2 = 9
  4. Multiply: 2 × 9 = 18 and -8 × 3 = -24
  5. Combine all terms: 18 - 24 + 6
  6. Simplify: 18 - 24 = -6, then -6 + 6 = 0

The answer is 0.

4 f(x) = 2x² - 7x + 11, find f(5) = ?

Hint: Substitute the given value into the function and carefully follow the order of operations (PEMDAS).

Show the answer

Answer: 26

  1. Start with the function f(x) = 2x² - 7x + 11
  2. Substitute x = 5 into the function: f(5) = 2(5)² - 7(5) + 11
  3. Calculate the exponent first: (5)² = 25
  4. Multiply: 2 × 25 = 50 and 7 × 5 = 35
  5. Substitute back: f(5) = 50 - 35 + 11
  6. Perform subtraction: 50 - 35 = 15
  7. Perform addition: 15 + 11 = 26

The answer is 26.

5 f(x) = 2x² - 5x + 15, find f(5) = ?

Hint: Substitute the given value into the function and follow the order of operations carefully. Remember to calculate exponents before multiplication, and multiplication before addition/subtraction.

Show the answer

Answer: 40

  1. Start with the function f(x) = 2x² - 5x + 15
  2. Substitute x = 5 into the function: f(5) = 2(5)² - 5(5) + 15
  3. Calculate the exponent first: (5)² = 25
  4. Multiply: 2 × 25 = 50 and 5 × 5 = 25
  5. Substitute back: f(5) = 50 - 25 + 15
  6. Perform subtraction and addition from left to right: 50 - 25 = 25, then 25 + 15 = 40

The answer is 40.

6 f(x) = 3x^2 - 5x + 2, find f(-2) = ?

Hint: Substitute the given value into the function and follow the order of operations carefully.

Show the answer

Answer: 24

  1. Start with the function f(x) = 3x^2 - 5x + 2
  2. Substitute x = -2 into the function: f(-2) = 3(-2)^2 - 5(-2) + 2
  3. Calculate the exponent first: (-2)^2 = 4
  4. Multiply: 3 × 4 = 12
  5. Multiply: -5 × (-2) = 10
  6. Add all terms: 12 + 10 + 2 = 24
  7. The final answer is 24
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