Course Cheatsheet
Section 03: Functions as Business Models
What is a Function?
A function is a rule that assigns to each input exactly one output.
| Notation | Meaning |
|---|---|
| \(f(x) = 2x + 5\) | Function \(f\) maps input \(x\) to output \(2x + 5\) |
| \(f(3) = 11\) | Substitute \(x = 3\): \(2(3) + 5 = 11\) |
| \(C(x), R(x), P(x)\) | Cost, Revenue, Profit functions of quantity \(x\) |
A graph represents a function if and only if every vertical line intersects it at most once. Pass: lines, parabolas, exponential curves. Fail: circles (\(x^2 + y^2 = 25\)), sideways parabolas (\(x = y^2\)). This ensures no input maps to two different outputs.
Domain and Range
- Domain: set of all valid inputs
- Range: set of all possible outputs
When finding the domain, check for these three restrictions:
- Denominators: cannot be zero – set denominator \(\neq 0\)
- Square roots: argument must be \(\geq 0\)
- Logarithms: argument must be \(> 0\)
- Business context: quantities \(\geq 0\), capacity limits, time constraints
If none of these apply (e.g., polynomials), the domain is all real numbers.
Domain Rules by Function Type
| Function Type | Restriction | Domain Rule |
|---|---|---|
| Rational: \(\frac{1}{x-a}\) | Denominator \(\neq 0\) | \(x \neq a\) |
| Square Root: \(\sqrt{x-a}\) | Argument \(\geq 0\) | \(x \geq a\) |
| Logarithm: \(\log(x-a)\) | Argument \(> 0\) | \(x > a\) |
| Polynomial | None | All real numbers |
Range Patterns
| Function Type | Range | Example |
|---|---|---|
| Linear \(f(x) = ax + b\) | \((-\infty, \infty)\) | All real numbers |
| Quadratic \(f(x) = x^2 + c\) | \([c, \infty)\) | Minimum at \(c\) |
| Square Root \(\sqrt{x}\) | \([0, \infty)\) | Non-negative |
| Rational \(\frac{1}{x}\) | \((-\infty, 0) \cup (0, \infty)\) | All except 0 |
Four Representations of Functions
- Verbal: “Base cost of 100 plus 3 per unit”
- Algebraic: \(C(x) = 100 + 3x\)
- Numerical (table): pairs of \((x, C(x))\) values
- Graphical: visual plot of the relationship
Piecewise Functions
When a function uses different rules for different input ranges:
\[f(x) = \begin{cases} 2x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}\]
- Evaluate by checking which condition \(x\) satisfies first
- \(f(-3) = 2(-3) + 1 = -5\) (use first rule since \(-3 < 0\))
- \(f(5) = 5^2 = 25\) (use second rule since \(5 \geq 0\))
Linear Functions
Forms of Linear Functions
Slope-Intercept Form: \(y = mx + b\)
- \(m\) = slope (rate of change per unit)
- \(b\) = y-intercept (value when \(x = 0\))
Point-Slope Form: \(y - y_1 = m(x - x_1)\)
- Use when you know one point and the slope
- Or compute slope from two points first: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
The slope \(m\) represents the marginal change – how much the output changes per one-unit increase in input.
- In a cost function \(C(x) = 500 + 25x\): slope = 25 means each additional unit costs 25 more
- In a demand function \(Q_d = 500 - 50p\): slope = \(-50\) means each 1 price increase loses 50 in demand
- Positive slope = increasing, negative slope = decreasing, zero = constant
Parallel and Perpendicular Lines
- Parallel: same slope (\(m_1 = m_2\)), e.g., two firms with same variable cost
- Perpendicular: \(m_1 \cdot m_2 = -1\)
Supply and Demand
Demand Functions
\(Q_d = a - bp\) (decreasing: higher price means lower quantity)
- \(a\): theoretical maximum quantity (at price 0)
- \(b\): price sensitivity
Supply Functions
\(Q_s = -c + dp\) (increasing: higher price means higher quantity)
Market Equilibrium
Set \(Q_d = Q_s\) and solve for equilibrium price \(p^*\), then find equilibrium quantity \(Q^*\).
Example: Demand \(Q_d = 500 - 50p\), Supply \(Q_s = -100 + 100p\)
\[500 - 50p = -100 + 100p \implies 600 = 150p \implies p^* = 4, \quad Q^* = 300\]
Cost-Volume-Profit (CVP) Analysis
| Component | Formula |
|---|---|
| Total Cost | \(TC = FC + VC \times Q\) |
| Revenue | \(R = P \times Q\) |
| Profit | \(\Pi = R - TC = (P - VC) \times Q - FC\) |
| Contribution Margin | \(CM = P - VC\) |
| Break-Even Quantity | \(Q_{BE} = \frac{FC}{CM} = \frac{FC}{P - VC}\) |
| Target Profit Quantity | \(Q = \frac{FC + \text{Target Profit}}{CM}\) |
| Margin of Safety | Actual sales \(-\) Break-even sales |
Linear Modeling from Data
Steps to create a linear model:
- Identify the independent and dependent variables
- Calculate slope between data points: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
- Use point-slope form to find the equation
- Interpret parameters in business context
Example: Sales increase by 15 units/month, starting at 120 units in month 1:
\[S(m) = 15m + 105\]
Check: \(S(1) = 15(1) + 105 = 120\) and \(S(2) = 135\).
Linear Depreciation
\[V(t) = V_0 - dt\]
- \(V_0\): initial value, \(d\): depreciation per period
- Useful life: \(n = V_0 / d\) periods
- Example: Vehicle at 30,000 depreciating 5,000/year: \(V(t) = 30{,}000 - 5{,}000t\), fully depreciated after 6 years
Quadratic Functions
Standard Form
\(f(x) = ax^2 + bx + c\)
- \(a > 0\): opens upward (U-shape, has minimum)
- \(a < 0\): opens downward (inverted U, has maximum)
- \(|a|\) larger means narrower parabola
- \(c\) = y-intercept
The Vertex Formula
\[x_v = -\frac{b}{2a}, \qquad y_v = f(x_v)\]
- If \(a < 0\): vertex is the maximum (profit optimization)
- If \(a > 0\): vertex is the minimum (cost optimization)
- Axis of symmetry: \(x = x_v\)
Vertex Form
\(f(x) = a(x - h)^2 + k\) where vertex is \((h, k)\)
Completing the Square
Convert \(f(x) = ax^2 + bx + c\) to vertex form:
- Factor out \(a\): \(a(x^2 + \frac{b}{a}x) + c\)
- Add/subtract \(\left(\frac{b}{2a}\right)^2\) inside
- Rewrite as \(a(x - h)^2 + k\)
Example: \(2x^2 - 12x + 10 = 2(x^2 - 6x + 9 - 9) + 10 = 2(x-3)^2 - 8\)
Revenue Optimization
When demand depends on price: \(Q = a - bp\)
- Revenue: \(R(p) = p \cdot Q = p(a - bp) = ap - bp^2\)
- Optimal price: \(p^* = \frac{a}{2b}\)
- Maximum revenue: \(R(p^*) = \frac{a^2}{4b}\)
These give different optimal prices! Revenue maximization ignores costs entirely. Profit \(\Pi(p) = R(p) - C(p)\) includes cost structure, shifting the optimum. Always check which you are asked to maximize.
Area Optimization
Classic problem: maximize area with a constraint.
Example: 200m of fencing, one side against a building:
- Constraint: \(2x + y = 200 \implies y = 200 - 2x\)
- Area: \(A(x) = x(200 - 2x) = 200x - 2x^2\)
- Maximum at: \(x = \frac{200}{4} = 50\)m, so dimensions are 50m x 100m = 5,000 m\(^2\)
Break-Even Points (Quadratic)
Solve \(P(x) = 0\) using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The profitable region lies between the two break-even points (when \(a < 0\)).
Function Transformations
Vertical Transformations
| Transformation | Formula | Business Example |
|---|---|---|
| Shift up by \(k\) | \(f(x) + k\) | Fixed cost increase |
| Shift down by \(k\) | \(f(x) - k\) | Government subsidy |
| Stretch by \(a > 1\) | \(a \cdot f(x)\) | Price markup |
| Compress by \(0 < a < 1\) | \(a \cdot f(x)\) | Discount |
| Reflect over x-axis | \(-f(x)\) | Revenue to loss |
Horizontal Transformations
| Transformation | Formula | Business Example |
|---|---|---|
| Shift right by \(h\) | \(f(x - h)\) | Market entry delay |
| Shift left by \(h\) | \(f(x + h)\) | Early launch |
| Compress (faster) | \(f(bx)\), \(b > 1\) | Accelerated lifecycle |
| Stretch (slower) | \(f(x/b)\), \(b > 1\) | Extended lifecycle |
| Reflect over y-axis | \(f(-x)\) | Reverse time |
Horizontal transformations work opposite to intuition:
- \(f(x - 3)\) shifts right by 3 (minus = right)
- \(f(x + 3)\) shifts left by 3 (plus = left)
- \(f(2x)\) compresses horizontally (narrower)
- \(f(x/2)\) stretches horizontally (wider)
Remember: inside changes act in reverse!
Combining Transformations
For \(g(x) = a \cdot f(b(x - h)) + k\), apply in order:
- Horizontal shift by \(h\)
- Horizontal scale by \(b\)
- Vertical scale by \(a\) (includes reflection if \(a < 0\))
- Vertical shift by \(k\)
Function Composition
Definition
\((f \circ g)(x) = f(g(x))\): apply \(g\) first, then \(f\) to the result.
Example (Supply Chain):
- Raw materials to components: \(g(x) = 2x + 10\)
- Components to products: \(f(y) = 3y + 50\)
- Total: \((f \circ g)(x) = 3(2x + 10) + 50 = 6x + 80\)
\((f \circ g)(x) \neq (g \circ f)(x)\) in general. For example, if \(f(x) = 2x + 3\) and \(g(x) = x^2 - 1\):
- \((f \circ g)(x) = 2(x^2 - 1) + 3 = 2x^2 + 1\)
- \((g \circ f)(x) = (2x + 3)^2 - 1 = 4x^2 + 12x + 8\)
Always apply the inner function first, then feed the result to the outer function.
Domain of Compositions
For \((f \circ g)(x)\): start with domain of \(g\), then ensure \(g(x)\) falls in the domain of \(f\).
Inverse Functions
Definition
If \(f(a) = b\), then \(f^{-1}(b) = a\). The inverse “undoes” the original function.
\(f^{-1}(x)\) is the inverse function (reverses the mapping). It is not the same as \(\frac{1}{f(x)}\) (the reciprocal).
- Inverse: \(f^{-1}\) satisfies \(f(f^{-1}(x)) = x\)
- Reciprocal: \(\frac{1}{f(x)}\) is just one divided by the output
Example: If \(f(x) = 2x\), then \(f^{-1}(x) = \frac{x}{2}\), but \(\frac{1}{f(x)} = \frac{1}{2x}\).
Finding Inverses (Step-by-Step)
- Replace \(f(x)\) with \(y\)
- Swap \(x\) and \(y\)
- Solve for \(y\)
- Replace \(y\) with \(f^{-1}(x)\)
- Verify: \(f(f^{-1}(x)) = x\)
Example: \(f(x) = 3x + 6\)
- \(y = 3x + 6 \to x = 3y + 6 \to y = \frac{x-6}{3}\)
- \(f^{-1}(x) = \frac{x-6}{3}\)
Invertibility
A function is invertible only if it is one-to-one (each output from exactly one input).
- Horizontal line test: every horizontal line hits the graph at most once
- Linear functions (non-constant) are always invertible
- Quadratic functions are not invertible without domain restriction
Graphical Property
The graph of \(f^{-1}\) is the reflection of \(f\) over the line \(y = x\).
- Domain of \(f\) = Range of \(f^{-1}\)
- Range of \(f\) = Domain of \(f^{-1}\)
Business Inverses
| Forward | Inverse | Use |
|---|---|---|
| \(Q = 1000 - 20p\) | \(p = \frac{1000 - Q}{20}\) | Price needed for target demand |
| \(C = 500 + 25x\) | \(x = \frac{C - 500}{25}\) | Units producible within budget |
| \(P(x) = 20x - 3000\) | \(x = \frac{P + 3000}{20}\) | Units needed for target profit |
When reading economic graphs, systematically identify:
- Intercepts: y-intercept = fixed costs or initial value; x-intercepts = break-even points
- Slope / Rate of change: marginal values and trends
- Maximum / Minimum: optimal points (vertex of parabola)
- Intersections: equilibrium points, where two functions are equal
- Shape: linear (constant rate), quadratic (optimization), exponential (growth)
- Profitable region: area between cost and revenue curves where \(R > C\)
- Vertical distance: at any \(x\), the gap between curves represents profit or loss
Key Vocabulary
| Term | Definition |
|---|---|
| Function | Rule assigning exactly one output to each input |
| Domain | Set of all valid input values |
| Range | Set of all possible output values |
| Slope | Rate of change \(m = \Delta y / \Delta x\) |
| Break-even | Where profit equals zero (\(R = C\)) |
| Contribution margin | Price minus variable cost (\(CM = P - VC\)) |
| Vertex | Maximum or minimum point of a parabola |
| Composition | Chaining functions: \((f \circ g)(x) = f(g(x))\) |
| Inverse | Function that reverses the original mapping |
| Equilibrium | Where supply equals demand |
Quick Reference
Key Formulas
| Formula | Purpose |
|---|---|
| \(m = \frac{y_2 - y_1}{x_2 - x_1}\) | Slope from two points |
| \(Q_{BE} = \frac{FC}{P - VC}\) | Break-even quantity |
| \(x_v = -\frac{b}{2a}\) | Vertex x-coordinate |
| \(p^* = \frac{a}{2b}\) | Optimal price (demand \(Q = a - bp\)) |
| \((f \circ g)(x) = f(g(x))\) | Composition |
| \(f^{-1}\): swap \(x, y\) and solve | Inverse |
Problem-Solving Strategies
- Identify function type: linear, quadratic, composed, inverse?
- Check domain: mathematical restrictions and business constraints
- Write the function: express the relationship algebraically
- Apply technique: vertex formula, composition, equilibrium, etc.
- Interpret: translate the mathematical answer back to business context
- Verify: does the answer make practical sense?
- Composition order: \((f \circ g)\) means apply \(g\) first, then \(f\) – not the other way around
- Inverse vs reciprocal: \(f^{-1}(x) \neq \frac{1}{f(x)}\)
- Horizontal shifts: \(f(x - 3)\) shifts right, not left
- Optimization: revenue-maximizing price \(\neq\) profit-maximizing price
- Domain neglect: always check for division by zero, negative square roots, and business constraints
- Units confusion: keep track of whether variables represent price, quantity, cost, or time