IB MYP 4-5 Maths- Domain and range- Study Notes - New Syllabus
IB MYP 4-5 Maths- Domain and range – Study Notes
Extended
- Domain and range
IB MYP 4-5 Maths- Domain and range – Study Notes – All topics
Domain and Range of Functions
Domain & Range
The domain of a function is the set of all possible input values (x-values) for which the function is defined. It answers the question: “What x-values can I put into this function without breaking any rules?”
The range of a function is the set of all possible output values (y-values) that the function can produce. It answers the question: “What y-values can the function output?”
Key Rules to Determine Domain:
- Square roots: The expression inside a square root must be ≥ 0.
- Fractions: The denominator cannot be 0.
- Logarithms: The argument (inside log) must be > 0.
- Trigonometric functions: Domain may exclude undefined values (e.g., tan x is undefined at \(90^\circ, 270^\circ,\dots\)).
Steps to Find Domain and Range:
- Start with all real numbers as possible inputs.
- Apply restrictions based on the type of function.
- Analyze graph behavior to confirm range.
Domain and Range from Graphs:
- Domain: Look at x-values covered by the graph (left to right).
- Range: Look at y-values covered by the graph (bottom to top).
Common Functions and Their Domain & Range:
| Function | Domain | Range |
|---|---|---|
| \( y = x \) (Linear) | \( (-\infty, \infty) \) | \( (-\infty, \infty) \) |
| \( y = x^2 \) (Quadratic) | \( (-\infty, \infty) \) | \( [0, \infty) \) |
| \( y = \sqrt{x} \) | \( [0, \infty) \) | \( [0, \infty) \) |
| \( y = \frac{1}{x} \) | \( x \neq 0 \) | \( (-\infty, 0) \cup (0, \infty) \) |
| \( y = \log x \) | \( (0, \infty) \) | \( (-\infty, \infty) \) |
| \( y = e^x \) | \( (-\infty, \infty) \) | \( (0, \infty) \) |
Example :
Find the domain and range of \( f(x) = \dfrac{2}{x-3} \).
▶️ Answer/Explanation
Step 1: Denominator cannot be zero → \( x-3 \neq 0 \) → \( x \neq 3 \).
Domain: All real numbers except 3: \( (-\infty, 3) \cup (3, \infty) \).
Range: Output never equals 0 → \( (-\infty, 0) \cup (0, \infty) \).
Example :
Find the domain and range of \( f(x) = \sqrt{x+2} \).
▶️ Answer/Explanation
Step 1: Inside square root ≥ 0 → \( x+2 \ge 0 \) → \( x \ge -2 \).
Domain: \( [-2, \infty) \).
Range: Square root is non-negative → \( [0, \infty) \).
Example :
A function models the height \( h(t) \) of water in a tank as \( h(t) = \sqrt{100 – t^2} \), where \( t \) is time in seconds. Find domain and range.
▶️ Answer/Explanation
Step 1: Inside square root ≥ 0 → \( 100 – t^2 \ge 0 \) → \( -10 \le t \le 10 \).
Domain: \( [-10, 10] \).
Range: Max height is 10 when t = 0 → \( [0, 10] \).