How to Solve: Domain and Range

How to Solve: Domain and Range

Introduction

"If you can find the domain and range of any function, you’ll unlock 5–10 marks on your exam—and avoid losing points on word problems about real-world limits, like how high a drone can fly or how much profit a business can make!

What You Need To Know First

1. What a function is: A rule that assigns exactly one output (y) for every input (x).
2. Inequalities: How to solve and graph x > 3, x ≤ -2, etc.
3. Graph basics: How to read and sketch linear, quadratic, and absolute-value graphs.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Identify the Type of Function

• Is it a polynomial (e.g., x² + 3x - 4)?
• A rational function (e.g., 1/(x-2))?
• A root function (e.g., √(x+5))?
• A piecewise function (e.g., f(x) = x² if x < 0, 2x if x ≥ 0)?

Step 2: Find the Domain

For polynomials (linear, quadratic, cubic, etc.): - Domain is all real numbers (-∞ < x < ∞).

For rational functions (fractions): 1. Set the denominator ≠ 0. 2. Solve for x. 3. Exclude those x-values from the domain.

For square roots: 1. Set the inside ≥ 0. 2. Solve for x. 3. Domain is all x that satisfy the inequality.

For logarithms: 1. Set the inside > 0. 2. Solve for x. 3. Domain is all x that satisfy the inequality.

For piecewise functions: - Find the domain for each piece, then combine them.

Step 3: Find the Range

Method 1: Use the Graph 1. Sketch or visualise the graph. 2. Look for the lowest and highest y-values. 3. Check for asymptotes (horizontal or vertical).

Method 2: Algebraic Approach - For quadratics (f(x) = ax² + bx + c): - If a > 0, range is [vertex y-value, ∞). - If a < 0, range is (-∞, vertex y-value]. - For linear functions (f(x) = mx + b): - Range is all real numbers (-∞ < y < ∞). - For rational functions: - Find horizontal asymptotes (if any) to determine limits. - For square roots: - Range starts at 0 and goes up (y ≥ 0).

Step 4: Write the Domain and Range in Correct Notation

• Interval notation: Use ( ) for "not included," [ ] for "included."
• Example: x > 2 → (2, ∞)
• Example: x ≤ 5 → (-∞, 5]
• Set notation: Use {x | x > 2}.
• Inequality notation: 2 < x < 5.

Step 5: Check for Restrictions (Word Problems)

• If the problem gives a real-world context, adjust domain/range accordingly.
• Example: "A ball is thrown upward from 5m high" → y ≥ 5.
• Example: "A company can’t produce negative items" → x ≥ 0.

Worked Examples

Example 1 - Basic: Polynomial Function

Problem: Find the domain and range of f(x) = x² - 4x + 3.

Step 1: Identify the function type → Quadratic polynomial. Step 2: Domain → All real numbers (-∞ < x < ∞). Step 3: Range → Since a = 1 > 0, parabola opens upward. - Find vertex: x = -b/(2a) = 4/2 = 2. - f(2) = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1. - Range: y ≥ -1 → (-1, ∞). Step 4: Write in interval notation. - Domain: (-∞, ∞) - Range: (-1, ∞)

What we did and why: - Polynomials always have a domain of all real numbers. - For quadratics, the range depends on the vertex and whether the parabola opens up or down.

Example 2 - Medium: Rational Function with Restrictions

Problem: Find the domain and range of f(x) = (x + 1)/(x - 3).

Step 1: Identify the function type → Rational function. Step 2: Domain → Denominator ≠ 0 → x - 3 ≠ 0 → x ≠ 3. - Domain: (-∞, 3) ∪ (3, ∞). Step 3: Range → Find horizontal asymptote. - Degree of numerator = degree of denominator → y = 1 (leading coefficients). - Check if y = 1 is ever reached: 1 = (x + 1)/(x - 3) → x - 3 = x + 1 → -3 = 1 (no solution). - Range: y ≠ 1 → (-∞, 1) ∪ (1, ∞). Step 4: Write in interval notation. - Domain: (-∞, 3) ∪ (3, ∞) - Range: (-∞, 1) ∪ (1, ∞)

What we did and why: - Rational functions are undefined where the denominator is zero. - Horizontal asymptotes help find the range, but we must check if the function ever equals the asymptote.

Example 3 - Exam Style: Piecewise Function with Word Problem

Problem: A drone is launched from the ground (y = 0). Its height in meters after t seconds is given by: - h(t) = -5t² + 20t for 0 ≤ t ≤ 4 - h(t) = 0 for t > 4

Find the domain and range of h(t).

Step 1: Identify the function type → Piecewise function. Step 2: Domain → Given by the problem: - 0 ≤ t ≤ 4 (drone is in the air) and t > 4 (drone has landed). - Domain: [0, ∞). Step 3: Range → Find max height of the quadratic part. - h(t) = -5t² + 20t (opens downward). - Vertex at t = -b/(2a) = -20/(2-5) = 2. - h(2) = -5(2)² + 20(2) = -20 + 40 = 20. - Range: 0 ≤ y ≤ 20 (since drone starts and ends at y = 0). Step 4: Write in interval notation. - Domain: [0, ∞) - Range: [0, 20]

What we did and why: - Piecewise functions require checking each part separately. - Word problems add restrictions (e.g., time can’t be negative, height can’t be negative).

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in for your exam. Domain is all the x-values your function can take. Range is all the y-values it can spit out. Here’s how to crush it:

1. For domain:
2. Polynomials? All real numbers.
3. Fractions? Denominator ≠ 0.
4. Square roots? Inside ≥ 0.
5. Logs? Inside > 0.

6. Piecewise? Check each piece.

7. For range:

8. Quadratics? Find the vertex.
9. Fractions? Look for horizontal asymptotes.
10. Square roots? Starts at 0 and goes up.

11. Word problems? Adjust for real-world limits.

12. Write it right:

13. Use interval notation: ( ) for not included, [ ] for included.
14. Watch for traps—examiners love hiding restrictions!

You’ve got this. Now go practice 3 problems tonight, and you’ll own domain and range on exam day!