Session 03 - Functions

Mathematics for Master Students

Author

Dr. Tobias Vlcek

Recap & Your Questions

Session 02 in a Nutshell

  • A set is an unordered collection of distinct elements: \(\in\) for elements, \(\subseteq\) for sets
  • \(\cup\), \(\cap\), \(\setminus\), \(^c\) mirror or, and, not: De Morgan flips \(\cup\) and \(\cap\)
  • Counting a union: \(|A \cup B| = |A| + |B| - |A \cap B|\)
  • \(A \times B\) collects ordered pairs: \(\mathbb{R} \times \mathbb{R} = \mathbb{R}^2\) is the plane
  • Constraints are sets, and they combine by intersection
  • Today: functions map between sets, and their graphs live in \(\mathbb{R}^2\)

Common Issues in the Homework

  • Element vs subset: braces build a set: \(\{2\} \subseteq A\) pairs with \(\subseteq\), while \(2 \in A\) pairs with \(\in\); mixing them was the most frequent slip
  • Complements reverse inclusion: from \(A \subseteq B\) it follows that \(B^c \subseteq A^c\): writing \(A^c \subseteq B^c\) is the classic inversion error
  • Mixed \(\cup\) and \(\cap\): \(A \cap B \cup C\) is ambiguous: grouping changes the set, so always add parentheses

. . .

Which tasks gave you trouble? Bring them up now: we take the time to go through them before we move on to today’s topic.

Warm-Up: One Quick Evaluation

NoteQuestion

Let \(A = \{1, 2, 3, 4\}\) and \(B = \{3, 4, 5\}\). Compute \(\;(A \cup B) \setminus (A \cap B)\).

. . .

  • Inside out: \(A \cup B = \{1, 2, 3, 4, 5\}\) and \(A \cap B = \{3, 4\}\)
  • Removing the intersection: \((A \cup B) \setminus (A \cap B) = \{1, 2, 5\}\)
  • In words: everything in exactly one of the two sets: “either, but not both”

Today’s Plan

  • What is a function?: exactly one output for every input
  • Function gallery: linear, quadratic, polynomial, rational, exponential, logarithm
  • Domain in practice: which inputs are allowed
  • Combining functions: composition and inverses
  • Properties: monotone, bounded, convex vs concave

Functions as Mappings

What Is a Function?

A function \(f: A \to B\) is a rule that assigns to each element \(x \in A\) exactly one element \(f(x) \in B\).

  • It maps between two sets: the ones from last session
  • Think of a machine: input \(x\) goes in, output \(f(x)\) comes out
  • “Exactly one” makes it predictable: same input, same output, every time

. . .

WarningCommon Mistake

“Exactly one” restricts the outputs per input. Two different inputs may share the same output: that is perfectly fine.

Function Notation

\[f: \mathbb{R} \to \mathbb{R}, \quad x \mapsto 2x + 5\]

  • \(f\) is the function’s name; \(x\) is the input, also called the argument
  • \(f(x)\) is the output: read “\(f\) of \(x\)”; here \(f(x) = 2x + 5\)
  • \(x \mapsto f(x)\) reads “\(x\) maps to \(f(x)\)”: note the little bar on the arrow
  • \(f: A \to B\) names the two sets: inputs come from \(A\), outputs land in \(B\)

. . .

\(f\) and \(f(x)\) are different things: \(f\) is the rule, \(f(x)\) is a number: the value at \(x\).

Domain, Codomain, Range

For \(f: A \to B\), three sets matter:

  • The domain \(A\): all allowed inputs
  • The codomain \(B\): where outputs are declared to land
  • The range \(f(A) = \{f(x) : x \in A\}\): the outputs that actually occur
  • Always \(f(A) \subseteq B\): the range is a subset of the codomain

. . .

Example: \(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = x^2\): codomain \(\mathbb{R}\), but range \([0, \infty)\): no square is negative.

Evaluating a Function

To evaluate, substitute the input everywhere the variable appears, take \(f(x) = x^2 - 2x\):

  • \(f(3) = 3^2 - 2 \cdot 3 = 9 - 6 = 3\)
  • \(f(0) = 0 - 0 = 0\)
  • \(f(-1) = (-1)^2 - 2 \cdot (-1) = 1 + 2 = 3\)
  • Note: \(f(3) = f(-1) = 3\): two inputs, same output, allowed

. . .

Put negative inputs in parentheses before squaring: \((-1)^2 = 1\), not \(-1\).

Plugging in Whole Expressions

The input can be an expression: substitute all of it, in parentheses. Again \(f(x) = x^2 - 2x\):

\[f(a + h) = (a + h)^2 - 2(a + h) = a^2 + 2ah + h^2 - 2a - 2h\]

. . .

WarningCommon Mistake

\(f(a + h) \neq f(a) + f(h)\), check: \(f(1 + 1) = f(2) = 0\), but \(f(1) + f(1) = -1 - 1 = -2\).

. . .

Expressions like \(f(a + h)\) are the raw material of Session 4: “how much does \(f\) change when the input moves by \(h\)?”

The Vertical Line Test

The graph of \(f\) is the set \(\{(x, f(x)) : x \in A\} \subseteq \mathbb{R}^2\): ordered pairs, as in Session 2.

  • A curve is the graph of a function \(\Leftrightarrow\) every vertical line hits it at most once
  • Why: one input \(x\) must yield exactly one output
  • The circle \(x^2 + y^2 = 1\) fails: at \(x = 0\) it contains two points, \(y = 1\) and \(y = -1\)
  • Lines, parabolas and exponential curves all pass

Your Turn: Which Graph is a Function?

NoteQuestion

Which of the two graphs shows \(y\) as a function of \(x\)?

. . .

Only Graph B: A fails the vertical line test: \(x = 1\) gives both \(y = 1\) and \(y = -1\). The kink in B is fine.

Functions in Business

Functions map decisions to outcomes, the core of every quantitative model:

  • Cost function \(C(q)\): production quantity \(\to\) total cost
  • Demand function \(D(p)\): price \(\to\) units sold
  • Production function: input (labor hours) \(\to\) output (units)
  • One decision in, one predicted outcome out: that is why models must be functions

. . .

Whenever a spreadsheet column is computed from another column, there is a function behind it.

Domain in Practice

The Maximal Domain

If no domain is stated, take the maximal domain: the largest set of reals on which the formula is defined.

Three red flags:

  • Division by zero: denominators must stay \(\neq 0\)
  • Even roots of negatives: under \(\sqrt{\phantom{x}}\) we need \(\geq 0\)
  • Logs of non-positives: inside \(\ln\) we need \(> 0\) (strictly!)

. . .

Everything else (polynomials, exponentials) is safe on all of \(\mathbb{R}\).

Finding the Domain: Examples

Function Restriction Maximal domain
\(\dfrac{1}{x - 3}\) \(x - 3 \neq 0\) \(\mathbb{R} \setminus \{3\}\)
\(\sqrt{2x + 6}\) \(2x + 6 \geq 0\) \([-3, \infty)\)
\(\ln(5 - x)\) \(5 - x > 0\) \((-\infty, 5)\)

. . .

The results are intervals. Session 1’s notation pays off: square bracket at \(-3\) (\(\sqrt{0}\) is fine), round bracket at \(5\) (\(\ln 0\) is not).

Your Turn: Find the Domain

NoteQuestion

What is the maximal domain of \(\;f(x) = \dfrac{\sqrt{x - 2}}{x - 4}\;\)?

. . .

  • The root needs \(x - 2 \geq 0\), so \(x \geq 2\)
  • The denominator needs \(x - 4 \neq 0\), so \(x \neq 4\)
  • Both must hold, intersect the conditions: \([2, 4) \cup (4, \infty)\)

. . .

Collect every restriction first, then combine: one forgotten condition is the typical exam slip.

Domains in Business

Business adds its own restrictions on top of the math:

  • Quantities are non-negative: \(q \in [0, \infty)\): no producing \(-50\) pallets
  • Capacity caps the allowed quantities: \(q \in [0, 500]\)
  • Prices must keep demand meaningful: \(D(p) = 1200 - 20p \geq 0\) forces \(p \in [0, 60]\)
  • Model domain = mathematical domain \(\cap\) business constraints: an intersection, as in Session 2

Combining Functions

Composition of Functions

Chaining two functions, apply \(g\) first, then \(f\) to the result:

\[(f \circ g)(x) = f(g(x))\]

  • Read from the inside out: the inner function acts first
  • Example: \(g(x) = 2x\) and \(f(x) = x + 3\): \(\;(f \circ g)(x) = f(2x) = 2x + 3\)
  • But \((g \circ f)(x) = g(x + 3) = 2x + 6\): not the same
  • Business reading: multi-stage processes, the output of one stage feeds the next

Order Matters: Voucher and VAT

A webshop applies a €10 voucher \(v(p) = p - 10\) and 19% VAT \(t(p) = 1.19p\) to a net price \(p\):

  • Voucher first, then tax: \((t \circ v)(p) = 1.19(p - 10) = 1.19p - 11.90\)
  • Tax first, then voucher: \((v \circ t)(p) = 1.19p - 10\)
  • The results differ by €1.90 on every order: for the customer, voucher first is cheaper

. . .

WarningCommon Mistake

Assuming \(f \circ g = g \circ f\): in general, the order cannot be swapped. Always check which function acts first.

Your Turn: Compose in the Right Order

NoteQuestion

Let \(f(x) = x^2\) and \(g(x) = x + 1\). Compute \(\;(f \circ g)(2)\;\) and \(\;(g \circ f)(2)\).

. . .

  • \((f \circ g)(2) = f(g(2)) = f(3) = 9\)
  • \((g \circ f)(2) = g(f(2)) = g(4) = 5\)
  • Inside out, always, and the order visibly changes the answer

Inverse Functions

The inverse \(f^{-1}\) undoes \(f\):

\[f(a) = b \quad \Leftrightarrow \quad f^{-1}(b) = a\]

  • It exists only if \(f\) is one-to-one: every output comes from exactly one input
  • Graph check: every horizontal line hits the graph at most once
  • \(f(x) = x^2\) on \(\mathbb{R}\) fails: \(f(2) = f(-2) = 4\): which input produced \(4\)?
  • Restricting to \([0, \infty)\) repairs it: there \(f^{-1}(x) = \sqrt{x}\)

Finding an Inverse

Write \(y = f(x)\), solve for \(x\), then read the result as a function of the output.

. . .

Demand example: \(q = D(p) = 1200 - 20p\):

\[q = 1200 - 20p \quad \Rightarrow \quad 20p = 1200 - q \quad \Rightarrow \quad p = 60 - \frac{q}{20}\]

. . .

  • \(D^{-1}(q) = 60 - \frac{q}{20}\): the price needed to sell exactly \(q\) units
  • Check with a pair: \(D(30) = 600\) and \(D^{-1}(600) = 60 - 30 = 30\) ✓
  • Domain and range swap: the domain of \(D^{-1}\) is the range of \(D\)

The Notation Trap: \(f^{-1} \neq 1/f\)

  • \(f^{-1}(x)\) is the inverse function: the \(-1\) sits on the function name
  • The reciprocal of the value is written \((f(x))^{-1} = \frac{1}{f(x)}\)
  • Example \(f(x) = 2x\): \(\;f^{-1}(x) = \frac{x}{2}\), but \(\frac{1}{f(x)} = \frac{1}{2x}\): completely different

. . .

WarningCommon Mistake

Reading \(f^{-1}\) as “one over \(f\)”: for functions, the exponent \(-1\) means undo, not divide.

The Mirror Property

. . .

Swapping input and output reflects the graph across \(y = x\): the point \((3, 4)\) on \(f\) becomes \((4, 3)\) on \(f^{-1}\): exactly how \(\ln x\) mirrored \(e^x\).

Properties

Monotonicity

\(f\) is increasing if \(x_1 < x_2 \Rightarrow f(x_1) \leq f(x_2)\): strictly increasing with \(<\). Decreasing works the same way, flipped.

  • Graph reading: strictly increasing = rises left to right, everywhere
  • Costs increase in quantity; demand decreases in price
  • Strictly monotone \(\Rightarrow\) one-to-one \(\Rightarrow\) invertible: that is why \(e^x\) has an inverse but \(x^2\) on \(\mathbb{R}\) does not

Boundedness

\(f\) is bounded above if some \(M\) satisfies \(f(x) \leq M\) for all \(x\): bounded below works with \(\geq m\).

  • \(e^x > 0\): bounded below by \(0\), unbounded above
  • A market share lives in \([0, 1]\): bounded on both sides
  • \(f(x) = x^2\): bounded below by \(0\), but grows without limit
  • Bounds tell you what a model can, and cannot, predict

Convex or Concave?

  • Convex: for two graph points, the chord lies above the graph: \(x^2\), \(e^x\)
  • Concave: the chord lies below the graph: \(\ln x\), \(\sqrt{x}\): diminishing returns
  • Intuition only for now: Sessions 4–5 make it precise and find optima

Closing

Key Takeaways

  • A function assigns each input exactly one output: domain, codomain and range are sets
  • Know the gallery shapes: linear, quadratic, polynomial, rational, exponential, logarithm
  • \(\ln x\) undoes \(e^x\): the power rule \(\ln(x^k) = k \ln x\) solves for exponents
  • Maximal domain: no division by zero, no even roots of negatives, no logs of non-positives
  • Composition chains processes: order matters; inverses undo, but only one-to-one functions have them
  • Monotone, bounded, convex/concave: reading behavior from a graph

Skip order if running long: 1. Mirror Property, 2. Compound Interest (keep one bullet on Growth & Decay), 3. Notation Trap (compress into Inverse Functions callout), 4. compress Functions in Business to its punchline.

That’s it for today.

Any Questions?

Until the Next Session

  • Work through the Tasks: problems with worked solutions
  • Check yourself with the Self-Test quiz
  • Keep the Cheatsheet next to you while practicing
  • Note down anything unclear: we start next session with your questions

. . .

When a function confuses you, sketch it: five plotted points reveal more than ten minutes of staring at the formula.

Preview: Session 04: Differentiation

  • Central question: how fast does a function change?
  • Slopes of curved graphs: the tangent line
  • Today’s \(f(a + h)\) becomes the difference quotient
  • Marginal cost and marginal revenue: the language of economic decisions
  • And \(e^x\) reveals its superpower

. . .

See you there, and bring your questions!

Literature & Further Reading

  • These sessions cover the essentials: textbooks offer more depth and practice
  • Sydsaeter & Hammond: Essential Mathematics for Economic Analysis
  • Jacques: Mathematics for Economics and Business
  • Full recommendations on the tutorial’s literature page