Teaching objectives
What is (and isn't) a function? Before working with formulas and specific families, it's worth understanding the idea of a correspondence itself: one element of the domain, exactly one image. Students will learn to tell functions apart from "non-functions" using finite sets and real graphs, meet the injective, surjective and bijective properties (including how to count them), and learn to find the domain of a real function, both from its graph and from its formula.
What you'll learn
Students build an intuitive understanding of a function as a correspondence, the counterexamples that break it, the three classic properties (injective, surjective, bijective) with their counts, and the procedure for finding the domain of a real function.
- Rooms 1–2: the concept of a function with finite sets (arrows), domain and range, and a comparison between two functions sharing a domain.
- Room 3: counterexamples — correspondences that are not functions (double arrows, reversed arrows, elements with no arrow).
- Room 4: the vertical line test for recognizing functions in real graphs.
- Rooms 5–8: injective, surjective and bijective, with recognition and counting (including the horizontal line test for injectivity in real graphs).
- Rooms 9–10: visual domain and range in real functions, read directly from the graph.
- Rooms 11–12: forbidden operations (dividing by 0, an even-index root of a negative, the logarithm of a non-positive number) and how they determine the algebraic domain, distinguishing open from closed intervals.
- Room 13: piecewise functions — the domain is the union of each piece's domain.
- Room 14: domain as an intersection of intervals, when two forbidden operations act at once.
- Room 15: final review of the whole lab.
Key mathematical ideas
- A function is a one-to-one correspondence: one x → exactly one f(x). If some x has two images, or none at all, it is not (or is not well defined as) a function.
- Domain: the set of values for which the function is defined. Range (image): the set of values f actually takes.
- A function is injective if distinct elements always map to distinct images; checked on a graph with the horizontal line test.
- A function is surjective if every element of the codomain is covered by some arrow; counting them sometimes requires subtracting by complement (the total minus those that fail the condition).
- A function is bijective if it is injective and surjective at once; between two sets of the same size n, the number of bijections is n!.
- The algebraic domain of a function is found by avoiding three forbidden operations: dividing by 0 (excludes a point), taking an even-index root of a negative number (excludes a closed interval, with a bracket), and taking the logarithm of a non-positive number (excludes an open interval, with a parenthesis).
- In a piecewise function, the total domain is the union of each piece's domain; when two conditions act simultaneously on the same formula, the domain is their intersection.
Room-by-room contents
Room 1 · Sets and arrows
The concept of a function is introduced through an arrow diagram between two sets: each element of the domain points to exactly one element of the codomain. A chained quiz reviews domain, codomain and range.
Student tasks
- Observe the arrow diagram and identify the domain, codomain, range, and why an element with no arrow falls outside the domain.
- Answer the 5 chained questions.
Room 2 · Two functions, the same domain
Two finite-to-finite correspondences with the same domain but different ranges, shown side by side for comparison.
Student tasks
- Work out the range of each function separately.
- Explain why they share a domain but not a range.
Room 3 · Is it a function?
Four correspondences between {A, B, C} and {1, 2, 3}, shown one at a time: a valid one, one with two arrows from the same origin, one with a reversed arrow, and one with an element that has no arrow.
Student tasks
- Decide in each of the 4 cases whether the correspondence is or is not a function.
- Identify the specific reason when it is not.
Room 4 · The vertical line test
Four mini-graphs (a circle, the parabola y = x², the sideways parabola x = y², the vertical line x = 1.5), each with an overlaid test vertical.
Student tasks
- Determine for each graph whether the test vertical crosses it more than once.
- Conclude whether each one is or is not a function.
Room 5 · Injective functions
Two example correspondences (one injective, one not) are revealed one at a time in arrow notation, each with its own question and its own hint if you slip up. Then a counting challenge: how many injective functions exist from a 2-element set to a 3-element set.
Student tasks
- Recognize whether each example is injective.
- Work out how many injective functions exist from {1,2} to {a,b,c}.
Room 6 · The horizontal line test
Four real function graphs (y = x², y = x³/3, y = |x|, y = 2x − 1) are revealed one at a time, each with a test horizontal, to decide whether they are injective.
Student tasks
- Determine for each graph whether the test horizontal crosses it more than once.
- Conclude whether each one is or is not injective.
Room 7 · Surjective functions
Two example correspondences are revealed one at a time, each with its own question and its own hint if you slip up. Then a tougher counting challenge: how many surjective functions exist from {a, b, c} to {0, 1}.
Student tasks
- Recognize whether each example is surjective.
- Work out how many surjective functions exist from {a,b,c} to {0,1} (by complement: 2³ − 2).
Room 8 · Bijective functions
Two example correspondences (bijective, or neither injective nor surjective) are revealed one at a time. Then a counting challenge via factorial: how many bijections exist between two 4-element sets.
Student tasks
- Recognize whether each example is bijective.
- Work out 4! and explain why it counts the bijections.
Room 9 · Exploring y = √(9 − x²)
The student moves an x-slider and sees in real time how f(x) changes on the graph of y = √(9 − x²) (a semicircle). Domain and image are read directly from the graph.
Student tasks
- Move x across its full range and observe how f(x) changes.
- Determine the domain and image of the function by reading the graph.
Room 10 · More domain and range
Two more explorers in sequence: g(x) = √(x + 4), whose domain is unbounded above, and h(x) = 4 − x², whose domain is ℝ but whose range is bounded above.
Student tasks
- Read the domain and range of g(x) = √(x + 4).
- Read the domain and range of h(x) = 4 − x².
Room 11 · Forbidden operations
The three operations that restrict the domain are introduced: dividing by 0, an even-index (square) root of a negative, and the logarithm of a non-positive number, with three worked examples. It stresses that even roots give closed intervals (brackets) and logarithms give open ones (parentheses).
Student tasks
- Work out the domain of f(x) = 3/(x − 2).
- Work out the domain of f(x) = √(x − 5).
- Work out the domain of f(x) = ln(x − 3).
Room 12 · Practice: domain with one forbidden operation
Four more domain exercises with a single forbidden operation: a simple division, a root, a logarithm, and a division with a quadratic denominator (two excluded points). The distractors deliberately mix open and closed intervals.
Student tasks
- Work out the domain of f(x) = 5/(x + 3).
- Work out the domain of g(x) = √(2x − 6).
- Work out the domain of h(x) = ln(4 − x).
- Work out the domain of k(x) = 4/(x² − 9).
Room 13 · Piecewise functions
f(x) = √(x+2) if x ≤ 1, x−1 if x>1 is presented. The student computes the domain of each piece separately and checks that the total domain is their union.
Student tasks
- Work out the domain of the first piece.
- Work out the domain of the second piece.
- Union both domains to get the total domain.
Room 14 · Domain as an intersection of intervals
f(x) = √(x − 1) + √(6 − x) is presented, with two simultaneous forbidden operations. The student works out each condition separately and intersects them.
Student tasks
- Work out the condition from the first radicand.
- Work out the condition from the second radicand.
- Intersect both conditions to get the total domain.
Room 15 · Final review
Five chained questions reviewing the whole lab: is it a function?, injective, surjective, bijective, and the domain of a real function.
Student tasks
- Answer the 5 review questions.
Rooms to project
The most striking ones to show and discuss in class.