Domain and Range

Grade 9 · algebra · 101 practice problems · read aloud

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What are Domain and Range? 🤔

The domain is the complete set of all possible input values (x-values) for a function. The range is the complete set of all possible output values (y-values). Think of a function like a vending machine: the domain is the money you can put in (inputs), and the range is the snacks you can get out (outputs). Understanding domain and range helps us know the limits of a function's behavior.

How to Find Domain and Range: A Step-by-Step Guide

  1. Look at the function type. Is it a line, a parabola, a square root?
  2. For Domain (x-values): Ask yourself, "What x-values CAN'T be used?"
    • You can't divide by zero.
    • You can't take the square root of a negative number.
  3. For Range (y-values): Ask yourself, "What y-values are possible from the outputs?"
    • Look at the graph's highest and lowest points.
    • Consider the function's behavior.
  4. Write your answer in inequality notation (e.g., x > 2) or interval notation (e.g., (2, ∞)).

Worked Examples

Example 1: Linear Function

Function: f(x) = 2x + 1

Step 1: This is a straight line. There are no restrictions.

Step 2 (Domain): You can plug any x-value in. Domain: All Real Numbers, or (-∞, ∞)

Step 3 (Range): The line goes infinitely up and down. Range: All Real Numbers, or (-∞, ∞)

Example 2: Quadratic Function

Function: f(x) = x²

Step 1: This is a parabola opening upwards with a vertex at (0,0).

Step 2 (Domain): You can square any number. Domain: (-∞, ∞)

Step 3 (Range): x² is always zero or positive. It never goes below the x-axis. Range: y ≥ 0, or [0, ∞)

Common Mistakes to Avoid 🚫

  • Swapping Domain and Range: Remember, Domain = Inputs (x), Range = Outputs (y). A good memory trick: D comes before R in the alphabet, just like X comes before Y.
  • Forgetting Restrictions: Always check for division by zero and square roots of negatives!
  • Misreading Graphs: Look carefully to see if the graph has a starting/ending point or holes.

Tips & Tricks

  • DIXI-ROYD: A great memory aid! Domain = X values = Independent variable. Range = Y values = Dependent variable.
  • Sketch it! Drawing a quick graph is the easiest way to visualize the range.
  • For lines that aren't horizontal, the domain and range are almost always "All Real Numbers."

How to Practice

To master domain and range:

  1. Start by identifying the function type from an equation.
  2. Practice with a mix of graphs and equations.
  3. Use online graphing tools (like Desmos) to type in an equation and instantly see its domain and range.
  4. Create a study sheet with common functions (linear, quadratic, square root) and their typical domain/range.

Practice problems

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

1 f(x) = x² - 4x + 3; f(5) = ?

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

Show the answer

Answer: 8

  1. Start with the function f(x) = x² - 4x + 3
  2. Substitute x = 5 into the function: f(5) = (5)² - 4(5) + 3
  3. Calculate the exponent: (5)² = 25
  4. Calculate the multiplication: -4(5) = -20
  5. Combine all terms: 25 - 20 + 3
  6. Perform the operations from left to right: 25 - 20 = 5, then 5 + 3 = 8

The answer is 8.

2 f(x) = 2x² - 8x + 6; 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 the order of operations.

Show the answer

Answer: 0

  1. Substitute x = 3 into the function. f(3) = 2*(3)² - 8*(3) + 6
  2. Calculate the exponent first. (3)² = 9 So f(3) = 2*9 - 8*3 + 6
  3. Perform the multiplications. 2*9 = 18 8*3 = 24 So f(3) = 18 - 24 + 6
  4. Perform the addition and subtraction from left to right. 18 - 24 = -6 -6 + 6 = 0 Therefore, f(3) = 0.

We are given the function f(x) = 2x² - 8x + 6 and asked to find f(3).

3 f(x) = 2x² - 4x + 1; f(3) = ?

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: 7

  1. Write the function with x replaced by 3. f(3) = 2*(3)² - 4*(3) + 1
  2. Evaluate the exponent first. (3)² = 9 So f(3) = 2*9 - 4*3 + 1
  3. Perform the multiplications. 2*9 = 18 4*3 = 12 So f(3) = 18 - 12 + 1
  4. Perform the addition and subtraction from left to right. 18 - 12 = 6 6 + 1 = 7 Therefore, f(3) = 7.

We are given the function: f(x) = 2x² - 4x + 1 We want to find f(3), which means we substitute x = 3 into the function.

4 f(x) = 2x² - 3x + 1, f(2) = ?

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

Show the answer

Answer: 3

  1. Write down the function. f(x) = 2x² - 3x + 1
  2. Substitute x = 2 into the function. This means everywhere you see x, replace it with 2. f(2) = 2*(2)² - 3*(2) + 1
  3. Follow the order of operations (PEMDAS/BODMAS). First, calculate the exponent: (2)² = 4. So f(2) = 2*4 - 3*2 + 1
  4. Perform the multiplications. 2*4 = 8 3*2 = 6 So f(2) = 8 - 6 + 1
  5. Perform the addition and subtraction from left to right. 8 - 6 = 2 2 + 1 = 3
  6. State the final answer. f(2) = 3 Thus,

We are given the function f(x) = 2x² - 3x + 1 and asked to find f(2). the correct answer is 3.

5 f(x) = 2x² - 5x + 3; f(4) = ?

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

Show the answer

Answer: 15

  1. Write the function: f(x) = 2x² - 5x + 3
  2. Substitute x = 4: f(4) = 2(4)² - 5(4) + 3
  3. Calculate the exponent: (4)² = 16
  4. Multiply: 2 × 16 = 32 and -5 × 4 = -20
  5. Add all terms: 32 - 20 + 3
  6. 32 - 20 = 12, then 12 + 3 = 15

The answer is 15.

6 f(x) = 4x² - 7x + 2; f(2) = ?

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

Show the answer

Answer: 4

  1. Start with the function f(x) = 4x² - 7x + 2
  2. Substitute x = 2 into the function: f(2) = 4(2)² - 7(2) + 2
  3. Calculate the exponent: (2)² = 4
  4. Multiply: 4 × 4 = 16 and -7 × 2 = -14
  5. Combine all terms: 16 - 14 + 2
  6. Simplify: 16 - 14 = 2, then 2 + 2 = 4

The answer is 4.

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