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.

Your Turn: What Is the Range?

NoteQuestion

\(f: \mathbb{R} \to \mathbb{R}\), \(f(x) = x^2 + 1\). Which set is the range of \(f\)?

(a) \(\mathbb{R}\) (b) \([0, \infty)\)
(c) \([1, \infty)\) (d) \((1, \infty)\)

. . .

(c): \(x^2 \geq 0\), so \(x^2 + 1 \geq 1\), and the value \(1\) is reached at \(x = 0\): square bracket.

. . .

  1. is the codomain, not the range: the declared landing zone is larger than the outputs that actually occur.

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 \(h\)?”

Your Turn: Evaluate

NoteQuestion

Again \(f(x) = x^2 - 2x\). What is \(f(-2)\)? Type the number. Bonus: simplify \(f(x + 1)\).

. . .

\[f(-2) = (-2)^2 - 2 \cdot (-2) = 4 + 4 = 8\]

. . .

\[f(x + 1) = (x + 1)^2 - 2(x + 1) = x^2 + 2x + 1 - 2x - 2 = x^2 - 1\]

. . .

WarningCommon Mistake

\(f(-2) = 0\) comes from dropping a sign: either \((-2)^2 = -4\) or \(-2 \cdot (-2) = -4\). Both minus signs matter.

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 (labour 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 maths:

  • 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 \(D\)

Your Turn: Undo the Function

NoteQuestion

Let \(f(x) = 3x - 6\). For which \(x\) is \(f(x) = 9\), i.e. what is \(f^{-1}(9)\)? Type the number.

. . .

  • Solve \(3x - 6 = 9\): \(\;3x = 15\), so \(x = 5\)
  • In general: \(y = 3x - 6 \Rightarrow x = \dfrac{y + 6}{3}\), hence \(f^{-1}(y) = \dfrac{y + 6}{3}\)
  • Check: \(f^{-1}(9) = \dfrac{15}{3} = 5\) ✓

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

Your Turn: Invertible or Not?

NoteQuestion

Which of these functions is not one-to-one on \(\mathbb{R}\), and therefore has no inverse there?

(a) \(f(x) = 2x + 1\) (b) \(f(x) = e^x\)
(c) \(f(x) = x^2 - 4\) (d) \(f(x) = x^3\)

. . .

(c): \(f(2) = f(-2) = 0\), one output from two inputs, so it fails the horizontal line test.

. . .

The other three are strictly increasing on all of \(\mathbb{R}\), and strictly monotone functions are always invertible.

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

Your Turn: Chord Check

NoteQuestion

Which of these functions is concave on its domain?

(a) \(x^2\) (b) \(e^x\)
(c) \(\sqrt{x}\) (d) \(\lvert x \rvert\)

. . .

(c): \(\sqrt{x}\) rises ever more slowly, its chords lie below the graph: diminishing returns.

. . .

  1. and (b) bend upward, and the chords of \(\lvert x \rvert\) also lie above its graph: all three are convex. Session 4 turns the chord test into a sign test on \(f''\).

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 behaviour 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 practising
  • 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
  • Start with Sydsæter et al. (2012) or Jacques (2015); full recommendations on the tutorial’s literature page
Jacques, Ian. 2015. Mathematics for Economics and Business. 8th ed. Always Learning. Pearson.
Sydsæter, Knut, Peter J. Hammond, and Arne Strøm. 2012. Essential Mathematics for Economic Analysis. 4th ed. Pearson.