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Professor Justin Lorieau

1. What Are Functions?

What Most Don't Get About Functions
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What Most Don't Get About Functions

⏱ 11:42YouTube 4K / 1080p✏️ Practice Problems β†—

1. What is a Function?

Section titled β€œ1. What is a Function?”

A function is a rule or mapping that takes one or more inputs and transforms them into one or more outputs. In physical sciences, this usually means taking one or more real numbers and generating a single output number.

f:X→Y,y=f(x)f: X \to Y, \quad y = f(x)
  • Independent variable (xx): The quantity you control or vary directly (e.g., temperature, time, or the number of reactants).
  • Dependent variable (yy): The quantity that responds to your inputs (e.g., reaction yield, kinetic energy, or pressure).

2. Multivariable Slices

Section titled β€œ2. Multivariable Slices”

When a function depends on multiple independent coordinates, such as z=f(x,y)z = f(x, y), the function describes a 2D surface living in 3D space:

z=f(x,y)=x2+y2z = f(x, y) = x^2 + y^2

In physical chemistry and quantum mechanics, we frequently examine 2D cross-sections by holding one variable constant (a partial derivative perspective):

(βˆ‚zβˆ‚x)y\left( \frac{\partial z}{\partial x} \right)_y

3. Domains of Validity: Classical vs. Relativistic Physics

Section titled β€œ3. Domains of Validity: Classical vs. Relativistic Physics”

A crucial error students make is assuming a scientific formula holds across all real numbers. Every physical equation has a strict domain of validity.

Consider the classical formula for kinetic energy:

Ek=12mv2E_k = \frac{1}{2} m v^2

In classical mechanics, vv can mathematically take any value in [0,∞)[0, \infty). However, in relativistic physics, velocity is physically bounded by the speed of light cc:

E=Ξ³mc2=mc21βˆ’v2c2E = \gamma m c^2 = \frac{m c^2}{\sqrt{1 - \frac{v^2}{c^2}}}

As v→cv \to c, the classical formula underpredicts the true energy by orders of magnitude. A function is only as valid as its domain!

Companion Practice Problem Set

Section titled β€œCompanion Practice Problem Set”

Test your understanding of functional domains and parametric representations with these companion problems.

Problem 1: Real Domain of the Unit Circle

Plot the function r2=x2+y2r^2 = x^2 + y^2 for a unit circle with r=1r = 1 in the real domain. What are the constraints on the independent variable xx?

πŸ’‘ View Hint
Reorganize the circle equation to isolate y, and examine what values under the square root yield real numbers.
πŸ“ Worked Solution & Video Walkthrough

To plot with yy as the dependent variable:

y=Β±r2βˆ’x2=Β±1βˆ’x2y = \pm \sqrt{r^2 - x^2} = \pm \sqrt{1 - x^2}

For yy to remain in the real numbers R\mathbb{R}, the radicand must be non-negative:

1βˆ’x2β‰₯0β€…β€ŠβŸΉβ€…β€Šx2≀1β€…β€ŠβŸΉβ€…β€Šβˆ’1≀x≀11 - x^2 \ge 0 \implies x^2 \le 1 \implies -1 \le x \le 1

For every input x∈(βˆ’1,1)x \in (-1, 1), there are two solutions (+y+y and βˆ’y-y), reflecting symmetry about the xx-axis.

Problem 2: Parametric Representation of a Circle

Express the unit circle (r=1r = 1) in parametric form using angle ΞΈ\theta as the independent variable.

πŸ’‘ View Hint
Recall trigonometric relations x = r \cos\theta and y = r \sin\theta.
πŸ“ Worked Solution & Video Walkthrough

By introducing the parameter θ∈[0,2Ο€]\theta \in [0, 2\pi]:

{x(θ)=cos⁑(θ)y(θ)=sin⁑(θ)\begin{cases} x(\theta) = \cos(\theta) \\ y(\theta) = \sin(\theta) \end{cases}

This parameterization converts a multi-valued relation into two single-valued functions, greatly simplifying integrals in polar coordinates.