Quantitative Reasoning

1015SCG

Lecture 5


Functions

What is a function? 🤔








Function Machine











Functions

What is a function? 🤔

A function is a rule that associates a unique output to each input.

Definition: A function assigns to each element of $X$ (set of numbers) exactly one element of $Y$ (also a set of numbers).

The set $X$ is called the domain of the function and the set $Y$ is called the range of the function.



Functions

$f(x) = 3x+2$

$f(3) = 3(3)+2$

$\quad\;\;\,\, =9 + 2$

$\quad\;\;\,\, =11$

$f(-5) = 3(-5)+2$

$\quad\;\;\,\, = -15+2$

$\quad\;\;\,\, = -13$




Functions

$f(x) = x^2-2x+4$

$f(-4) = (-4)^2-2(-4)+4$

$\qquad\,\, =16 + 8+4$

$\qquad\,\,=28$

$f(2a) = (2a)^2-2(2a)+4$

$\qquad\,\,= 4a^2-4a+4$

 




Functions

The domain of $f(x)$ is the set of all possible values of $x$ for which the function is defined (i.e. all the values of $x$ that can be used with the function).

The range of $f (x)$ is the set of all possible values that can be returned by the function.



Functions

Consider the function $f(x)= \dfrac{1}{x-5}$



$f(0) = \dfrac{1}{(0)-5}$ $=-\dfrac{1}{5}$

$f(-2) = \dfrac{1}{(-2)-5}$ $=-\dfrac{1}{7}$

$f(9) = \dfrac{1}{(9)-5}$ $=\dfrac{1}{4}$

$f(5) = \,$ Not possible! 1/0!

Domain: All real values of $x,$ except $x=5.$

Range: All real values except $0.$




Elementary functions










Polynomials

Exponentials & Logarithms

Recall: Index Laws

  • Multiplying: \( a^m \times a^n = a^{m+n} \)
  • Dividing: \( \dfrac{a^m}{a^n} = a^{m-n} \)
  • Power of a power: \( \left(a^m\right)^n = a^{m \times n} \)
  • Power of a product: \( \left(ab\right)^n = a^n b^n \)
  • Zero index: \( a^0 = 1 \quad (a \neq 0) \)
  • Negative index: \( a^{-n} = \dfrac{1}{a^n} \)
Extra:
$\large a^{\frac{1}{n}} = \sqrt[n]{a}$

Recall: Logarithm Laws

  • Product Law: $\;\log_a\left(M\times N\right)$ $=\log_a M + \log_a N$
  • Quotient Law: $\;\log_a\left(\dfrac{M}{N}\right)$ $=\log_a M - \log_a N$
  • Power Law: $\;\log_a\left(N^p\right)$ $= p \times \log_a (N)$
  • Trivial identities: $\;\log_a(a) = 1\;$ and $\;\log_a(1) = 0 $

These rules work for any base \( a > 0 ,\) \( a \ne 1 .\)


Recall: Change of Base Rule

\[ \large \log_b (N) = \frac{\log_a (N)}{\log_a (b)} \] where \( a \) can be 10 (common log) or \( e \) (natural log).

Example: $\log_2 10$ $ =\dfrac{\log_{10} 10}{\log_{10} 2} $ $ \approx \dfrac{1}{0.3010}$ $\approx 3.32$




Trigonometric functions


Recall: sin / cos / tan   ratios

\(\sin\left(\theta \right) = \dfrac{\text{opposite}}{\text{hypotenuse}}\)

\(\cos\left(\theta \right) = \dfrac{\text{adjacent}}{\text{hypotenuse}}\)

\(\tan\left(\theta \right) = \dfrac{\text{opposite}}{\text{adjacent}}\)


Elementary functions
Function Expression
Polynomials (linear, quadratic, etc...) \(a_nx^n + a_{n-1}x^{n-1}+\cdots a_1 x + a_0\)
Absolute value \(|x|\)
Square root \(\sqrt{x}\)
Exponentials \(e^x,\; b^x\)
Logarithms \(\ln x, \;\log_b(x)\)
Trigonometric \(\sin x, \cos x, \tan x\)
Inverse Trigonometric \(\arcsin x, \arccos x, \arctan x\)
Hyperbolic* \(\sinh x, \cosh x, \tanh x\)
Inverse Hyperbolic* \(\text{arcsinh}\, x, \text{arccosh}\, x, \text{arctanh}\, x\)

* We won't use these functions in this course.



Where functions are used?

🤔

🌡️ Temperature Conversion

Convert Celsius to Fahrenheit:

\[ F(C) = \frac{9}{5}C + 32 \]

  • Linear function
  • Slope: \( \dfrac{9}{5} \)
  • Y-intercept: \( 32 \)

Used in weather reports, lab experiments, etc.


📊 Linear Regression

Predicting height from age: Suppose we collect data from children aged 2 to 13 and record their heights.


📊 Linear Regression

Predicting height from age:

The data points suggest a linear trend:

\[ h(a) = 6.58a + 70.36 \]

📈 Linear Regression

Predicting height from age:

The data points suggest a linear trend:

\[ h(a) = 6.58a + 70.36 \]

  • \( a \): age in years
  • \( h(a) \): predicted height in cm
  • Slope \( 6.58 \): average growth per year

This line is the line of best fit — found using linear regression.



Polynomial Regression

$y = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n + \epsilon$

We look for a least squares polynomial function of best fit.


🔬 Hooke's Law (Spring Force)

Force needed to stretch a spring:

\[ F(x) = -kx \]

  • Linear function
  • \( x \): displacement in meters
  • \( k \): spring constant (stiffness)

Used in physics, biomechanics, engineering.

We consider the differential equation: \(\ds m \frac{d^2 x}{dt^2}+k x = 0.\)


🔬 Hooke's Law (Spring Force)

The undamped spring

Use mouse to drag mass and release.

$x(t) = A \sin \left(\omega t \right)$

Source code


🔬 Hooke's Law (Spring Force)

Damped spring. Use mouse to drag mass and release.

Source code



☕️ Law of Cooling & 🧫 Population Growth
\(T(t)=\left(T_0-T_m\right) e^{-kt} + T_m\) $\ds P(t) = \frac{\theta P_0 e^{rt}}{\theta-P_0+P_0e^{rt}}$


💊 Medicine in the Body

Drug concentration decreases over time:

\[ C(t) = C_0 e^{-kt} \]

  • Exponential decay
  • \( C_0 \): initial concentration
  • \( k \): decay constant

Models how the body metabolizes medicine.



💊 Medicine in the Body

\[ C(t) = 100 e^{-0.3t} \]

This is a classic example of exponential decay, useful in pharmacology for understanding how a drug's concentration diminishes after administration.


📚 Diminishing Returns in Learning

Modelling learning over time:

Early learning is fast, then progress slows:

\[L(t) = 20 \ln(t + 1)\]

  • $t$: time spent learning (hours)
  • $L(t)$: performance level
  • Logarithmic growth: quick gains at first, then slower improvement

This model captures the idea of diminishing returns in real learning scenarios.



📚 Diminishing Returns in Learning

\[ L(t) = 20 \ln(t + 1) \]

This model shows how learning improves quickly at first and then slows over time, a classic example of diminishing returns in skill acquisition.


Trigonometric functions: Procedural Landscape Generation

\[ f(\mathbf{x}) = \sum_{i=0}^{N-1} A_i \cdot \big[\cos\left(2\pi \, \mathbf{k}_i \cdot \mathbf{x} + \phi_i\right) + \sin\left(2\pi \, \mathbf{k}_i \cdot \mathbf{x} + \theta_i\right)\big] \]


   Combined with Linear Algebra
to visualise the 3D surface! 🤯

Open live demo


Computer Graphics & Art


$f(x) = \sqrt{6^2-x^2}$

$g(x) = 2 + 2 \sin(\text{floor}(x-t) 4321)$

$h_k(x) = \dfrac{-5}{k}+\dfrac{2}{5}, $

$\qquad k=0,1,\ldots, 10$

Graphtoy


Computer Graphics & Art

$R =13 + 3\left(\dfrac{1}{2} + \dfrac{1}{2} \sin \left(2\pi t + \dfrac{z}{3}\right)\right)^4$

$y = z -\abs{x}\sqrt{ \dfrac{20 - \abs{x}}{35} }$

$x^{2}+y^{2}+z^{2}=R$


Open in Desmos




Art with functions in Excel

Try it yourself! 🌼😊

Tutorial:
Painting with Maths in Google Sheets
by Iñigo Quilez


Found Functions

Photos by Nikki Graziano


Time in a Bottle by Jim Croce


📝 Practice 😃

📝 Practice: Logarithms

Find without the calculator:

  • \(\log(1000)\)
  • \(\ln \left(e^3\right)\)
  • \(10^{\log(3)}\)
  • \(e^{\ln(4)}\)
  • \(\log(25)+ \log(4)\)


📝 Practice: Exponential functions

The 🦠 bacteria population is described by the following equation:

\(P(t) = 500 \times 10 ^{0.2 t}\)

$t$ is measured in years.

  1. Write this as an exponent with base $e$.
  2. Is the population growing or shrinking?
  3. What will be the population of bacteria after 3 years?
  4. After how long does it double or halve? - pick one based on answers to part 1.
  5. When would it reach a population of 200? of 2000? -pick one based on answers to part 1.

Solution part 1: Write with base e

\(P(t) = 500 \times 10 ^{0.2 t}\)

We rewrite:

\( 10^{0.2t} = \exp\left( \ln\left(10^{0.2 t} \right) \right) \) \( = \exp\big( 0.2 t\ln\left(10 \right) \big) \) \( = e^{0.2t \ln(10)} \)

So \(\; P(t) = 500\, e^{(0.2\ln 10)\, t} \)

Since \(0.2\ln (10) \approx 0.4605\):

\( P(t) \approx 500\, e^{0.4605 t} \)



Solution part 2: Growing or shrinking?

\(P(t) = 500 \times 10 ^{0.2 t}\)

The exponent coefficient is positive:

\(0.2 > 0\;\; \) \(\Rightarrow \;\; \) population grows.

So the bacteria population is growing.





Solution part 3: Population after 3 years

\(P(t) = 500 \times 10 ^{0.2 t}\)

Compute:

\( P(3) = 500 \times 10^{0.2 (3)} \) \( = 500 \times 10^{0.6} \)

\(10^{0.6} \approx 3.981\)

\[ P(3) \approx 500 \times 3.981 = 1990.5 \]

So after 3 years: \(\approx 1991\) bacteria



Solution part 4: Doubling time

Here we need to solve: $ 500 \times 10^{0.2t} = 1000 $

Divide both sides by 500:\(\; 10^{0.2t} = 2 \)

Take log base 10:

\[ 0.2t = \log_{10}(2) \]

\( \Ra \;t = \dfrac{\log_{10}(2)}{0.2} \) \(\approx \dfrac{0.3010}{0.2} \) \( = 1.505 \)

Doubling time ≈ 1.51 years


Solution part 5: When does it reach a chosen population?

Consider population = $2000.$ Solve: $500 \times 10^{0.2t} = 2000 $

Divide both sides by 500:\(\; 10^{0.2t} = 4 \)

Take log base 10: \(\;0.2t = \log_{10}(4) = 0.6021 \)

\[ t = \frac{0.6021}{0.2} = 3.01 \]

So the population reaches 2000 after ≈ 3.0 years.



That's all for today!

See you in Week 6!