Theory
The crash your calculator refuses
Type 1 ÷ 0 into any calculator: error. Ask for √(−4) in real numbers: error again.
Every function has inputs it simply cannot accept. The set of inputs it CAN accept is its domain, and finding it is one of the most repeated exam questions in this unit. The outputs it actually produces form the range. Today you learn to find both, fast.
Theory
A bus route
Think of a function as a bus route. The domain is every stop where passengers may board. The codomain is every stop printed on the route map. The range is the stops where the bus actually drops someone today. The bus may never drop anyone at some printed stops, which is exactly range ⊊ codomain.
Theory
Finding a domain: hunt the forbidden values
For real-valued formulas, the domain is all real numbers EXCEPT values that break one of three rules:
- a denominator becomes 0
- an expression under a square root (or any even root) goes negative
- the argument of a logarithm is not positive
Example: f(x) = 1/(x − 3) breaks rule one at x = 3, so the domain is all reals except 3.
Quiz
What is the domain of f(x) = √(x − 2) over the real numbers?
- All real numbers
- All x with x ≥ 2
- All x with x > 2
- All x except 2
Show the answer
All x with x ≥ 2
The expression under the root must satisfy x − 2 ≥ 0, so x ≥ 2. Note that x = 2 is fine: √0 = 0 exists. Option C wrongly excludes 2 (that strictness belongs to logarithms and denominators), and option D applies the division rule to a root problem.
Theory
Finding a range: what actually comes out
The range asks: which y values are actually produced?
Two standard routes:
- Algebra: recognise built-in limits. y = x² can never be negative, so its range is y ≥ 0.
- Graph: the range is the vertical spread of the curve, every height the graph reaches.
Always state the range as a set or inequality, not as a vague description.
Think first
f(x) = x² + 5, with domain all real numbers.
Decide the range in your head before tapping. What is the smallest output the machine can make?
Show the answer
Range = all y ≥ 5.
x² is at least 0, so x² + 5 is at least 5, and every value above 5 is reached by some x. The +5 lifts the whole range up: a shift you can reuse on any "x² plus a constant" exam question without recomputing.
Theory
Three types: one-one, onto, bijective
Functions are classified by how inputs and outputs pair up:
- Injective (one-one): different inputs always give different outputs. Formally, f(a₁) = f(a₂) forces a₁ = a₂.
- Surjective (onto): every codomain element gets hit; range = codomain.
- Bijective: both at once, a perfect pairing with nothing doubled and nothing missed.
Example: f(x) = x + 1 on ℤ is bijective. f(x) = x² on ℝ is neither: f(2) = f(−2) kills one-one, and negative targets are never hit.
Quiz
f: ℝ → ℝ, f(x) = x². Why is it NOT surjective (onto)?
- Because f(2) and f(−2) are equal
- Because negative numbers in the codomain are never outputs
- Because its domain excludes negative numbers
- Because it is not a function at all
Show the answer
Because negative numbers in the codomain are never outputs
Onto asks: is every codomain element reached? Since no x gives x² = −1, the answer is no. Option A is a true fact, but it disproves ONE-ONE, not onto; keeping the two definitions separate is exactly what examiners test. The domain does include negatives, and f is a perfectly valid function.
Watch out
The three mix-ups that cost marks
Writing range where the question says codomain (they differ unless the function is onto). Excluding the boundary in root domains (x ≥ 2 includes 2; only division and log force strict inequality). And using the f(2) = f(−2) argument against onto-ness, when it disproves one-one. Slow down at these three moments.
Theory
Where you will meet this again
Input validation in code IS domain thinking: reject the values that crash the formula. Hash functions are studied by how far from one-one they are, and a bijection is why encryption can be undone by exactly one key. Next topic: building function formulas from tables and graphs.
Summary
Key takeaways
- Domain = allowed inputs; exclude division by zero, negatives under even roots, non-positive log arguments.
- Range = outputs actually produced; read it from algebraic limits or the graph's vertical spread.
- Range ⊆ codomain, with equality exactly when the function is onto.
- One-one: no two inputs share an output. Onto: no codomain element is missed. Bijective: both.
- Memory hook: domain is the guest list, range is who actually came.