Function Notation

Grade 9 · algebra · 101 practice problems · read aloud

🔊 Listen to this explanation

Function Notation: f(x)

📝 What is Function Notation?

Function notation, like f(x), is a special way to name and write functions. Think of it as a machine: you put an input (x) into the function machine (f), and it gives you an output (f(x)). It's useful because it clearly shows the relationship between inputs and outputs and helps us work with multiple functions at once.

🛠️ How to Use Function Notation: A Step-by-Step Guide

  1. Identify the function and the input value.
  2. Substitute the input value for every x in the function's rule.
  3. Simplify the expression using the order of operations.

📚 Worked Examples

Example 1: If f(x) = 2x + 5, find f(3).

  1. Substitute 3 for x: f(3) = 2(3) + 5
  2. Simplify: f(3) = 6 + 5
  3. Final Answer: f(3) = 11

Example 2: If g(x) = x² - 4, find g(-2).

  1. Substitute -2 for x: g(-2) = (-2)² - 4
  2. Simplify: g(-2) = 4 - 4
  3. Final Answer: g(-2) = 0

⚠️ Common Mistakes to Avoid

  • f(x) does not mean "f times x." It is read as "f of x" and means the function f evaluated at x.
  • Forgetting parentheses with negatives: When substituting a negative number, always use parentheses. g(-2) means (-2)², not -2².
  • Misapplying the order of operations: Always do exponents before multiplication, and multiplication before addition/subtraction.

💡 Tips & Tricks

  • Think "Input" and "Output": The number inside the parentheses is your input. The entire f(x) is the output.
  • Use a T-table: Create a table with x-values and their corresponding f(x) values to see the pattern.
  • Remember PEMDAS: The order of operations is your best friend when simplifying!

🎯 How to Practice

To master function notation, try these activities:

  • Create flashcards with functions on one side and evaluated outputs on the other.
  • Practice with a partner—one person gives an input, the other calculates the output.
  • Find the outputs for multiple inputs (like f(1), f(2), f(3)) and plot the points on a graph to see the function visually.

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 f(x) = 2x² - 3x + 5, f(4) = ?

Hint: To evaluate a function at a specific value, substitute the given number for the variable in the function's expression and follow the order of operations.

Show the answer

Answer: 25

  1. Substitute x = 4 into the function. f(4) = 2*(4)² - 3*(4) + 5
  2. Evaluate the exponent first. (4)² = 16 So f(4) = 2*16 - 3*4 + 5
  3. Perform the multiplications. 2*16 = 32 3*4 = 12 So f(4) = 32 - 12 + 5
  4. Perform the addition and subtraction from left to right. 32 - 12 = 20 20 + 5 = 25 Thus, f(4) = 25.

We are given the function f(x) = 2x² - 3x + 5 and asked to find f(4).

2 f(x) = 2x² - 5x + 3, f(3) = ?

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

Show the answer

Answer: 6

  1. Start with the function f(x) = 2x² - 5x + 3
  2. Substitute x = 3 into the function: f(3) = 2(3)² - 5(3) + 3
  3. Calculate the exponent first: (3)² = 9
  4. Multiply: 2 × 9 = 18 and -5 × 3 = -15
  5. Combine all terms: 18 - 15 + 3
  6. Calculate from left to right: 18 - 15 = 3, then 3 + 3 = 6

The answer is 6.

3 f(x) = 4x² - 3x + 7, f(3) = ?

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

Show the answer

Answer: 34

  1. Start with the function f(x) = 4x² - 3x + 7
  2. Substitute x = 3 into the function: f(3) = 4(3)² - 3(3) + 7
  3. Calculate the exponent first: (3)² = 9
  4. Multiply: 4 × 9 = 36 and -3 × 3 = -9
  5. Combine the terms: 36 - 9 + 7
  6. Perform the operations from left to right: 36 - 9 = 27, then 27 + 7 = 34

The answer is 34.

4 f(x) = 2x² + 5x - 3, f(4) = ?

Hint: Substitute the given value into the function and follow the order of operations: exponents first, then multiplication, then addition and subtraction.

Show the answer

Answer: 49

  1. Write the function: f(x) = 2x² + 5x - 3
  2. Substitute x = 4 into the function: f(4) = 2(4)² + 5(4) - 3
  3. Calculate the exponent: (4)² = 16
  4. Perform multiplication: 2 × 16 = 32 and 5 × 4 = 20
  5. Substitute the results: f(4) = 32 + 20 - 3
  6. Perform addition and subtraction from left to right: 32 + 20 = 52, then 52 - 3 = 49

The answer is 49.

5 f(x) = 4x² - 3x + 8, f(3) = ?

Hint: Remember to substitute the given value into the function and follow the order of operations carefully.

Show the answer

Answer: 35

  1. Start with the function f(x) = 4x² - 3x + 8
  2. Substitute x = 3 into the function: f(3) = 4(3)² - 3(3) + 8
  3. Calculate the exponent first: (3)² = 9
  4. Multiply: 4 × 9 = 36 and -3 × 3 = -9
  5. Combine all terms: 36 - 9 + 8
  6. Perform the operations from left to right: 36 - 9 = 27, then 27 + 8 = 35

The answer is 35.

6 f(x) = 5x² - 7x + 9, f(8) = ?

Hint: Substitute the given value for x into the function and carefully follow the order of operations: exponents first, then multiplication, then addition and subtraction.

Show the answer

Answer: 273

  1. Start with the function f(x) = 5x² - 7x + 9
  2. Substitute x = 8 into the function: f(8) = 5(8)² - 7(8) + 9
  3. Calculate the exponent first: (8)² = 64
  4. Multiply: 5 × 64 = 320 and -7 × 8 = -56
  5. Combine all terms: 320 - 56 + 9
  6. Perform the operations from left to right: 320 - 56 = 264, then 264 + 9 = 273

The answer is 273.

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