Mathematics: Applications and Interpretation - Functions Study Notes

Hello future mathematical modelers! This chapter on Functions is absolutely central to the IB AI course. Functions are the backbone of mathematical modeling—they allow us to precisely describe how one quantity changes in relation to another in the real world (like how quickly a population grows or how a company's profit changes with production). Don't worry if some concepts seem abstract; we will break them down using clear analogies and focus heavily on interpretation and technology use, as is required in AI.

Key Takeaway for AI: In Applications and Interpretation, we focus on what the function means, how to use technology to analyze it, and how to apply it to a real-world context.

1. Defining and Understanding Functions

1.1 What is a Function?

A function is essentially a rule that assigns exactly one output value (y) to every input value (x). Think of it as a precise machine:

  • You put something in (the input, \(x\)).
  • The machine follows a rule (the function, \(f\)).
  • It spits out exactly one result (the output, \(y\)).

Analogy: Imagine a vending machine. If you push button A1, you always get a specific drink. If you push A1 and sometimes get a chocolate bar and sometimes get juice, it's not a function!

1.2 Function Notation

We use the notation \(y = f(x)\), which is read as "y equals f of x."

  • \(x\) is the independent variable (what you choose to input).
  • \(y\) or \(f(x)\) is the dependent variable (the result depends on \(x\)).

Example: If \(f(x) = x^2 + 3\), then \(f(4)\) means we substitute 4 for x: \(f(4) = (4)^2 + 3 = 19\).

1.3 Domain and Range

Understanding the limits of a function is crucial, especially when modeling real-world constraints.

a) Domain (Inputs)

The domain is the complete set of possible input values (\(x\)-values) for which the function is defined. In real-world modeling, the domain might be limited by common sense (e.g., time must be positive, \(t \ge 0\)).

Restrictions to Watch Out For:

  1. You cannot divide by zero. If \(f(x) = 1/x\), then \(x \ne 0\).
  2. You cannot take the square root (or any even root) of a negative number (in the real number system). If \(f(x) = \sqrt{x-5}\), then \(x-5 \ge 0\), so \(x \ge 5\).

b) Range (Outputs)

The range is the complete set of output values (\(y\)-values) that the function can produce. The range is often easiest to determine by looking at the graph using your GDC.

Quick Review Box: Vertical Line Test
To check if a graph represents a function, use the Vertical Line Test. If any vertical line intersects the graph at more than one point, it is NOT a function.

2. Graphical Features and Interpretation (GDC Skills Essential)

In AI, we rely heavily on graphing technology (GDC) to analyze functions quickly.

2.1 Intercepts

a) \(y\)-intercept: Where the graph crosses the \(y\)-axis. This happens when \(x = 0\).
Interpretation: In modeling, the \(y\)-intercept usually represents the initial value or starting condition.

b) \(x\)-intercepts (Roots or Zeros): Where the graph crosses the \(x\)-axis. This happens when \(f(x) = 0\).
GDC Skill: Use the 'Zero' or 'Root' function on your GDC to find these points accurately.

2.2 Asymptotes

An asymptote is a line that the function approaches closer and closer to, but never actually touches. They represent limitations in the real-world model.

  • Vertical Asymptotes (VA): Occur when \(x\) approaches a value where the function is undefined (often resulting in division by zero). Example: \(f(x) = 1/(x-3)\) has a VA at \(x=3\).
  • Horizontal Asymptotes (HA): Describe the behavior of the function as \(x\) approaches very large positive or very large negative values (\(x \rightarrow \pm \infty\)). Example: The maximum capacity of a population model.

Did you know? An exponential growth curve often has a horizontal asymptote representing zero (the starting population cannot go below zero), while an exponential decay curve might have an HA representing the minimum concentration achievable in a chemical reaction.

2.3 Increasing and Decreasing Intervals

A function is increasing if the graph rises as you move from left to right, and decreasing if the graph falls.

  • The points where the function changes from increasing to decreasing (or vice versa) are called local maximums and local minimums (collectively, extrema).
  • GDC Skill: Use the 'Maximum' or 'Minimum' feature on your GDC to find the coordinates of these extrema.
  • Interpretation: A local maximum might represent the peak profit or the highest temperature reached in a cycle.

3. Combining Functions: Composite and Inverse

3.1 Composite Functions

A composite function is created when the output of one function becomes the input of another. This is denoted as \(f(g(x))\) or \((f \circ g)(x)\).

Mnemonic: Think of the functions being applied in order from right to left: start with \(x\), apply \(g\), then apply \(f\) to the result.

Example: If \(f(x) = x^2\) and \(g(x) = x+1\):
\(f(g(x)) = f(x+1) = (x+1)^2\).
\(g(f(x)) = g(x^2) = x^2 + 1\).
Notice that order matters! \(f(g(x)) \ne g(f(x))\).

3.2 Inverse Functions

The inverse function, denoted \(f^{-1}(x)\), undoes the operation of the original function \(f(x)\). If \(f\) maps \(a\) to \(b\), then \(f^{-1}\) maps \(b\) back to \(a\).

Key Property: If you apply a function and then its inverse (or vice versa), you get the original input back: \(f(f^{-1}(x)) = x\).

How to find \(f^{-1}(x)\) (Step-by-step):

  1. Start with the equation \(y = f(x)\).
  2. Swap the positions of \(x\) and \(y\).
  3. Solve the new equation for \(y\).
  4. Replace \(y\) with \(f^{-1}(x)\).

Graphical Connection: The graph of \(f^{-1}(x)\) is a reflection of the graph of \(f(x)\) across the line \(y = x\).

Crucial Warning: The notation \(f^{-1}(x)\) means the inverse function. It does not mean the reciprocal (\(1/f(x)\)). This is a very common mistake!

Accessibility Note: One-to-One Functions
For an inverse function \(f^{-1}(x)\) to exist, the original function \(f(x)\) must pass the Horizontal Line Test. This means every output (y) must come from only one input (x). If a function fails this test (like a quadratic function), you must restrict the domain of \(f(x)\) before an inverse can be found.

4. Essential Function Types for Modeling

The AI syllabus requires comfort with applying and interpreting several specific types of functions for modeling real-world data.

4.1 Linear Functions: \(f(x) = mx + c\)

Linear models represent situations where the rate of change is constant.

  • \(m\) is the gradient (slope): the change in \(y\) divided by the change in \(x\).
  • \(c\) is the \(y\)-intercept (the initial value).
Real-world use: Calculating costs based on distance driven, or converting temperature scales.

4.2 Quadratic Functions: \(f(x) = ax^2 + bx + c\)

Quadratic models create a parabolic shape, useful for optimization and trajectory.

  • The sign of \(a\) determines the shape (if \(a>0\), opens up $\cup$; if \(a<0\), opens down $\cap$).
  • The vertex (\(x = -b/(2a)\)) gives the maximum or minimum value.
Real-world use: Modeling the trajectory of a thrown object, or maximizing profit in a business scenario.

4.3 Exponential Functions: \(f(x) = a b^x\) or \(f(x) = a e^{kx}\)

Exponential functions model rapid growth or decay where the rate of change is proportional to the current amount.

  • If \(b > 1\) or \(k > 0\): Growth (e.g., population growth, compound interest).
  • If \(0 < b < 1\) or \(k < 0\): Decay (e.g., radioactive decay, depreciation).
  • They typically have a Horizontal Asymptote at \(y=0\).

4.4 Logarithmic Functions: \(f(x) = \log_b (x)\)

Logarithmic functions are the inverse of exponential functions. They are used to compress large scales.

  • They have a Vertical Asymptote at \(x=0\).
  • They grow very slowly.
Real-world use: Measuring sound intensity (decibels), earthquake strength (Richter scale), or acidity (pH scale).

4.5 Power Functions: \(f(x) = ax^k\)

Power functions include simple polynomials and models where quantities relate via an exponent.

  • If \(k\) is a positive integer (\(x, x^2, x^3\)): standard polynomials.
  • If \(k\) is negative (\(x^{-1} = 1/x\)): reciprocal functions (rational).
  • If \(k\) is a fraction (\(x^{1/2} = \sqrt{x}\)): root functions.
Real-world use: Scaling laws in biology, or inverse square laws (where \(k=-2\)).

4.6 Periodic (Trigonometric) Functions

Periodic functions (like sine and cosine) model cyclical patterns.
General form for AI modeling: \(f(x) = a \sin(b(x-c)) + d\) or \(f(x) = a \cos(b(x-c)) + d\).

  • Amplitude (\(a\)): Half the distance between the maximum and minimum values.
  • Period: The length of one complete cycle (\(2\pi/b\) or \(360^\circ/b\)).
  • Vertical Shift (\(d\)): The mean value or equilibrium line.
  • Phase Shift (\(c\)): The horizontal shift of the starting point.
Real-world use: Modeling ocean tides, seasonal temperature variations, or biorhythms.

5. Transformations of Functions

Understanding transformations allows you to predict how changing the equation changes the graph, helping you fit models to data.

We start with a parent function \(y = f(x)\) and transform it to \(y = A f(B(x-C)) + D\).

5.1 Vertical Transformations (Affecting Output - Outside the function)

These changes are intuitive (what you see is what you get).

  • Vertical Translation (Shift): \(y = f(x) + D\)
    Moves the graph up by \(D\) units (if \(D>0\)) or down (if \(D<0\)).
  • Vertical Stretch/Compression: \(y = A f(x)\)
    Stretches the graph vertically by factor \(|A|\). If \(A\) is negative, it also reflects the graph across the \(x\)-axis.
5.2 Horizontal Transformations (Affecting Input - Inside the function)

These changes are counter-intuitive (they do the opposite of what the sign suggests).

  • Horizontal Translation (Shift): \(y = f(x-C)\)
    Moves the graph right by \(C\) units (if \(C>0\)) or left (if \(C<0\)).
  • Horizontal Stretch/Compression: \(y = f(Bx)\)
    Compresses the graph horizontally by factor \(1/|B|\). If \(B\) is negative, it also reflects the graph across the \(y\)-axis.

Mnemonic for Transformations:
Outside changes (A, D) affect Y (Vertical) and are Intuitive.
Inside changes (B, C) affect X (Horizontal) and are Counter-intuitive.

Step-by-Step Transformation Order:

When multiple transformations occur, follow this order:

  1. Reflections and Stretches/Compressions (A and B).
  2. Translations (C and D).

Use mapping notation to track points; it is often the clearest method:

\((x, y) \rightarrow (\frac{1}{B}x + C, Ay + D)\)

Key Takeaway for Functions
Functions are essential modeling tools. Focus your energy on interpreting the meaning of key features (intercepts, extrema, asymptotes) within a real-world context, and rely on your GDC for accurate calculation and visualization. Practice fitting the right function type (Linear, Quadratic, Exponential, etc.) to given data sets.