Difficulty: Intermediate | Prerequisites: Basic algebra (solving for a variable, plotting coordinates).
This is the mathematical toolkit you need before the rest of ACE 300 makes sense. The course uses functions to model how consumers and firms behave, graphs to visualise supply and demand, and systems of equations to find equilibrium prices and quantities. If you are comfortable rearranging equations and reading a Cartesian plane, you are ready.
A function maps every input to exactly one output. In economics you will use functions to express demand and supply, graph them using slope-intercept form or point-plotting, and solve systems of two equations (usually supply = demand) to find equilibrium. Master these mechanics and the rest of ACE 300 sits on solid ground.
Function
An association that maps each member x of a set X to a single member y of another set Y, written f: X → Y.
In simple terms, every input gives you exactly one output. If one input could give two different outputs, it is an association but not a function.
Domain
The set of all permissible input values (X) for a function.
Think of it as the full range of x-values you are allowed to plug in.
Range (codomain image)
The set of all output values (Y) that the function actually produces.
Think of it as every y-value the function can spit out.
Direct function
A function that maps price to quantity: Q(P).
In simple terms, you feed in a price and get back the quantity demanded or supplied at that price.
Inverse function
A function obtained by solving the direct function for price: P(Q).
Think of it as flipping the question: instead of "what quantity at this price?" you ask "what price at this quantity?"
Injective (one-to-one)
No two elements in the domain map to the same element in the range. Passes the horizontal line test.
In simple terms, every output came from one and only one input.
Surjective (onto)
Every element in the codomain is the image of at least one element in the domain.
In simple terms, nothing in the expected output set goes unused.
Slope-intercept form
y = mx + b, where m is the slope and b is the vertical intercept.
Think of it as the quickest way to read a linear equation: the coefficient on x tells you the steepness, and the constant tells you where the line crosses the vertical axis.
System of linear equations
Two or more linear equations considered together; a solution is a set of values that satisfies all equations at once.
In simple terms, you are looking for the point where all the lines cross.
A function f: X → Y assigns every element x in the domain X to exactly one element y in the range Y.
If a single input maps to two different outputs, the mapping is an association but not a function.
Set notation: x ∈ X means "x is an element of X."
We write y = f(x) to say "y is the image of x under f."
Example: f(x) = 5x + 4, x ∈ ℝ.
f(−1) = 5(−1) + 4 = −1
f(0) = 5(0) + 4 = 4
f(1) = 5(1) + 4 = 9
Each x-value produces one and only one y-value, tracing a straight line on the plane.
Functions describe consumer and firm behaviour: utility functions, profit functions, cost functions, demand functions.
A common mapping is from prices to quantities (f: P → Q) or vice versa.
Direct function: maps price to quantity, Q_D(P). "Give me a price, I will tell you the quantity demanded."
Inverse function: maps quantity to price, P(Q_D). "Give me a quantity, I will tell you the price."
Subscripts D (demand) and S (supply) distinguish the two sides of the market.
Q_D(P) = 10 – (1/5)P
At P = 0: Q_D = 10
At P = 50: Q_D = 0
Domain: [0, 50]. Range: [0, 10].
Start with Q_D = 10 – (1/5)P. Rearrange to isolate P:
(1/5)P = 10 – Q_D
P = 50 – 5Q_D
In function notation: P(Q_D) = 50 – 5Q_D
The inverse of f: X → Y is itself a function only when f is both:
Injective (one-to-one): no two inputs share the same output. Check with the horizontal line test.
Surjective (onto): every element of Y is hit by some element of X (the codomain equals the range).
A function that is both injective and surjective is called bijective, and its inverse is guaranteed to be a proper function.
The most reliable way to graph any function: pick several input values, compute the output, and plot each (input, output) pair.
For P(Q_D) = 50 – 5Q_D:
Q_D = 0 → P = 50 → coordinate (0, 50)
Q_D = 1 → P = 45 → coordinate (1, 45)
Q_D = 5 → P = 25 → coordinate (5, 25)
Q_D = 10 → P = 0 → coordinate (10, 0)
Because the function is linear, two points are enough; connect them with a straight line and extend through the full domain.
y = mx + b
m = slope = rise / run = (y₂ – y₁) / (x₂ – x₁)
b = vertical intercept (the value of y when x = 0)
For the inverse demand P(Q_D) = 50 – 5Q_D:
m = −5, b = 50.
Verify: (45 – 50) / (1 – 0) = −5/1 = −5. Checks out.
Plot the vertical intercept (0, b). Here (0, 50).
From that point, use the slope to find the next point. A slope of −5 means "move right 1, move down 5" → (1, 45).
Connect the two points with a straight line.
Vertical intercept: set the independent variable to zero and solve for the dependent variable.
P = 50 – 5(0) = 50 → coordinate (0, 50).
Horizontal intercept: set the dependent variable to zero and solve for the independent variable.
0 = 50 – 5Q_D → Q_D = 10 → coordinate (10, 0).
Suppose inverse demand is P(Q_D) = 50 – 5Q_D and inverse supply is P(Q_S) = 5Q_S.
Demand: use (0, 50) and (10, 0). Downward-sloping line.
Supply: vertical and horizontal intercept both at (0, 0). Use the equilibrium intersection as the second point to draw an upward-sloping line.
The two lines cross at the equilibrium, which you find by solving the system of equations (see next section).
Two or more linear equations considered at the same time.
A solution is the set of values that makes every equation true simultaneously.
For a unique solution, the number of independent equations must equal the number of unknowns.
Fewer equations than variables: the system is under-determined (infinitely many solutions or none).
More independent equations than variables: the system is over-determined (typically no solution).
System: (1) P = 50 – 5Q and (2) P = 5Q.
At the intersection, P(Q_D) = P(Q_S) and Q_D = Q_S, so we drop subscripts.
Equation (1) is already solved for P.
Substitute into equation (2): since both expressions equal P, set them equal to each other: 50 – 5Q = 5Q.
Solve for Q: 50 = 10Q → Q = 5.
Plug Q back into either equation to find P: P = 5(5) = 25.
Equilibrium: Q = 5, P = 25.
System: (1) 2x₁ + 2x₂ = 12 and (2) −3x₁ + x₂ = 6.
Solve (1) for x₂: x₂ = 6 – x₁.
Substitute into (2): −3x₁ + (6 – x₁) = 6 → −4x₁ + 6 = 6 → −4x₁ = 0 → x₁ = 0.
Back-substitute: x₂ = 6 – 0 = 6.
Solution: x₁ = 0, x₂ = 6.
Slope-intercept form: y = mx + b
Slope formula: m = (y₂ – y₁) / (x₂ – x₁)
Direct demand (example): Q_D(P) = 10 – (1/5)P
Inverse demand (example): P(Q_D) = 50 – 5Q_D
Inverse supply (example): P(Q_S) = 5Q_S
Vertical intercept: set the independent variable to 0, solve for the dependent variable.
Horizontal intercept: set the dependent variable to 0, solve for the independent variable.
System-solving procedure: isolate one variable in one equation, substitute into the other, solve, then back-substitute.
Demand and supply functions are how economists model markets for everything from corn futures to ride-share pricing. When a city sets a price ceiling on rent, the maths is the same: plug the ceiling price into the demand and supply functions, compare quantities, and the gap is the shortage.
Students often confuse the direct demand function Q(P) with the inverse demand function P(Q). In economics, graphs conventionally put P on the vertical axis and Q on the horizontal axis, so the curve you draw corresponds to the inverse form, P(Q), even though many textbooks introduce Q(P) first.
A horizontal line (e.g. y = 25 for all x) is a function, but its inverse is not a function because it fails the horizontal line test.
When solving a system by substitution, students sometimes substitute back into the same equation they rearranged rather than into the other equation. Both routes give the right answer, but substituting into the simpler equation reduces arithmetic mistakes.
Slope is rise over run, not run over rise. Mixing the two flips the sign or magnitude of m.
⚠️ You will be asked to convert between direct and inverse demand (and supply) forms. Practise the algebra until it is automatic.
⚠️ Solving supply = demand for equilibrium is one of the most frequently tested skills in the first half of ACE 300.
⚠️ Know how to read intercepts from a graph and from an equation. Exam questions often ask for the "choke price" (the vertical intercept of inverse demand) or the "maximum quantity" (the horizontal intercept).
⚠️ Graphing conventions in economics differ from pure maths: price on the vertical axis, quantity on the horizontal axis. Do not swap them.
True or False: A function can map two different inputs to the same output.
Fill in the blank: The slope of P(Q_D) = 50 – 5Q_D is ______.
True or False: To find the horizontal intercept, you set the independent variable to zero.
Fill in the blank: If P = 50 – 5Q and P = 5Q, then the equilibrium quantity Q = ______.
True or False: An inverse function exists whenever the original function is injective and surjective.
Answers: 1. True (allowed, as long as each input gives only one output). 2. −5. 3. False (you set the dependent variable to zero). 4. 5. 5. True.
Q: Given Q_D(P) = 20 – 2P, find the inverse demand function P(Q_D).
A: Q_D = 20 – 2P → 2P = 20 – Q_D → P = 10 – (1/2)Q_D. So P(Q_D) = 10 – 0.5Q_D.
Q: What are the vertical and horizontal intercepts of P(Q_D) = 10 – 0.5Q_D?
A: Vertical intercept: set Q_D = 0, P = 10 → (0, 10). Horizontal intercept: set P = 0, Q_D = 20 → (20, 0).
Q: Solve the system P = 10 – 0.5Q and P = Q.
A: Set equal: 10 – 0.5Q = Q → 10 = 1.5Q → Q = 20/3 ≈ 6.67. Then P = Q ≈ 6.67.
Q: A function maps x₁ to y₁ and x₂ to y₁. Is it still a function? Is it injective?
A: It is still a function (each input maps to one output). It is not injective, because two distinct inputs share the same output.
This material connects directly to consumer theory (utility functions and budget constraints use the same algebra), producer theory (cost and profit functions), and market equilibrium analysis throughout ACE 300. The graphing skills here are the foundation for welfare analysis: consumer surplus and producer surplus are areas on the supply-and-demand diagram you learn to draw in this section.
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