Properties of Functions - Formulas & Concepts
IB Mathematics Analysis & Approaches (SL & HL)
📐 Domain and Range
Definitions:
Domain: The set of all possible input values (x-values) for which the function is defined
Range: The set of all possible output values (y-values) that the function can produce
Notation:
\[f: A \to B\]
where \(A\) is the domain and \(B\) is the codomain (the range is a subset of \(B\))
Common Domain Restrictions:
• Division: Denominator cannot be zero: \(x \neq a\) where \(f(x) = \frac{1}{x-a}\)
• Square root (even roots): Expression under root must be non-negative: \(x \geq 0\) for \(\sqrt{x}\)
• Logarithms: Argument must be positive: \(x > 0\) for \(\log(x)\)
🔢 Types of Functions
One-to-One (Injective) Function:
Each element in the range corresponds to exactly one element in the domain.
\[\text{If } f(a) = f(b), \text{ then } a = b\]
Test: Horizontal Line Test - any horizontal line intersects the graph at most once
Onto (Surjective) Function:
Every element in the codomain is mapped to by at least one element in the domain. Range = Codomain.
For every \(b \in B\), there exists \(a \in A\) such that \(f(a) = b\)
Bijective Function:
A function that is both one-to-one AND onto
Only bijective functions have inverse functions
🔄 Composite Functions
Definition:
A composite function applies one function to the result of another function.
\[(f \circ g)(x) = f(g(x))\]
Read as "f composed with g" or "f of g of x"
Important Properties:
\[(f \circ g)(x) \neq (g \circ f)(x)\]
Composition is NOT commutative (order matters)
\[(f \circ (g \circ h))(x) = ((f \circ g) \circ h)(x)\]
Composition IS associative
Domain of Composite Function:
The domain of \(f \circ g\) consists of all \(x\) in the domain of \(g\) such that \(g(x)\) is in the domain of \(f\)
🔀 Inverse Functions
Definition:
The inverse function \(f^{-1}\) "undoes" the operation of function \(f\).
\[f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x\]
\[(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x\]
Finding Inverse Functions:
Step 1: Replace \(f(x)\) with \(y\)
Step 2: Swap \(x\) and \(y\)
Step 3: Solve for \(y\)
Step 4: Replace \(y\) with \(f^{-1}(x)\)
Important Properties:
• Domain of \(f^{-1}\) = Range of \(f\)
• Range of \(f^{-1}\) = Domain of \(f\)
• Graph of \(f^{-1}\) is the reflection of \(f\) in the line \(y = x\)
⚖️ Even and Odd Functions
Even Function:
A function is even if it has reflective symmetry about the y-axis.
\[f(-x) = f(x) \quad \text{for all } x\]
Examples: \(f(x) = x^2\), \(f(x) = \cos(x)\), \(f(x) = |x|\)
Odd Function:
A function is odd if it has rotational symmetry (180°) about the origin.
\[f(-x) = -f(x) \quad \text{for all } x\]
Examples: \(f(x) = x^3\), \(f(x) = \sin(x)\), \(f(x) = x\)
Properties:
• Even + Even = Even
• Odd + Odd = Odd
• Even × Even = Even
• Odd × Odd = Even
• Even × Odd = Odd
🔧 Function Transformations
Vertical Translation:
\[y = f(x) + k\]
Shifts graph up by \(k\) units (if \(k > 0\)) or down by \(|k|\) units (if \(k < 0\))
Horizontal Translation:
\[y = f(x - h)\]
Shifts graph right by \(h\) units (if \(h > 0\)) or left by \(|h|\) units (if \(h < 0\))
Vertical Stretch/Compression:
\[y = af(x)\]
Stretches vertically by factor \(|a|\) if \(|a| > 1\), compresses if \(0 < |a| < 1\)
Horizontal Stretch/Compression:
\[y = f(bx)\]
Compresses horizontally by factor \(\frac{1}{|b|}\) if \(|b| > 1\), stretches if \(0 < |b| < 1\)
Reflection in x-axis:
\[y = -f(x)\]
Reflection in y-axis:
\[y = f(-x)\]
📏 Line Tests
Vertical Line Test:
Determines if a graph represents a function.
If any vertical line intersects the graph at more than one point, it is NOT a function
Horizontal Line Test:
Determines if a function is one-to-one (has an inverse).
If any horizontal line intersects the graph at more than one point, it is NOT one-to-one
📉 Asymptotes
Vertical Asymptote:
\[x = a\]
Occurs where the function is undefined (typically where denominator = 0)
Horizontal Asymptote:
\[y = b\]
Describes the behavior as \(x \to \pm\infty\): \(\lim_{x \to \infty} f(x) = b\)
Oblique (Slant) Asymptote:
\[y = mx + c\]
Occurs when degree of numerator is exactly 1 more than degree of denominator
🧩 Piecewise Functions
General Form:
\[f(x) = \begin{cases} f_1(x) & \text{if } x \in D_1 \\ f_2(x) & \text{if } x \in D_2 \\ f_3(x) & \text{if } x \in D_3 \end{cases}\]
Important Concepts:
• Check continuity at boundary points
• Domain restrictions must not overlap
• Evaluate each piece in its specific interval
🔄 Periodic Functions
Definition:
A function is periodic if it repeats its values at regular intervals.
\[f(x + p) = f(x) \quad \text{for all } x\]
where \(p\) is the period (the smallest positive value for which this holds)
Common Examples:
• \(\sin(x)\) and \(\cos(x)\) have period \(2\pi\)
• \(\tan(x)\) has period \(\pi\)
• For \(f(x) = \sin(bx)\) or \(f(x) = \cos(bx)\), period = \(\frac{2\pi}{|b|}\)
📈 Monotonic Functions
Increasing Function:
\[\text{If } x_1 < x_2, \text{ then } f(x_1) \leq f(x_2)\]
Strictly increasing if \(f(x_1) < f(x_2)\)
Decreasing Function:
\[\text{If } x_1 < x_2, \text{ then } f(x_1) \geq f(x_2)\]
Strictly decreasing if \(f(x_1) > f(x_2)\)
💡 Quick Tip: Always check domain restrictions first, then analyze the function's behavior. Use your GDC to visualize transformations and verify properties graphically. Remember that understanding function properties helps in sketching graphs and solving equations efficiently.