Sample Notes: Functions

1.1 Understanding Functions and Related Terminology

Function

• A function is a rule or relationship that assigns exactly one output for each input in its domain.
• Mathematically: If f is a function from set A to set B, then every element x in A has exactly one image f(x) in B.
• Example: f(x) = 2x + 3 is a function because each value of x gives one unique output.

Domain and Range (Image Set)

• Domain: The set of all possible inputs (x-values) for which the function is defined.
• Range: The set of all possible outputs (f(x)-values) the function can produce.
• Example: For f(x) = √x, domain is x ≥ 0 and range is f(x) ≥ 0.

One–One and Many–One Functions

• One–One Function (Injective): Every output is mapped from only one unique input.
  • Test: Horizontal Line Test — if no horizontal line cuts the graph more than once.
  • Example: f(x) = x + 5 is one–one.
• Many–One Function: More than one input can give the same output.
  • Example: f(x) = x^2 is many–one because f(2) = 4 and f(–2) = 4.

Inverse Function

• The inverse of a function reverses the mapping: f⁻¹(y) = x if f(x) = y.
• Only one–one functions have an inverse that is also a function.
• Notation: f⁻¹(x).
• To find inverse:
  1. Replace f(x) with y
  2. Solve for x
  3. Interchange x and y
  4. Replace y with f⁻¹(x)

Composition of Functions

• Combining two functions: fg(x) = f(g(x)).
• Order matters: fg(x) ≠ gf(x) in general.
• Domain of fg is the set of x for which x ∈ Domain of g and g(x) ∈ Domain of f.

1.2 Finding the Domain and Range of Functions

Domain

• Identify restrictions in expressions like:
  • Denominator ≠ 0
  • Even roots must be ≥ 0 (e.g., √x needs x ≥ 0)
• Example: f(x) = 1/(x – 3) → Domain: x ≠ 3

Range

• Use inverse method or sketching graphs.
• Consider all valid output values after applying domain restrictions.

Inverse and Composite Domains

• Domain of f⁻¹ is the range of f
• Domain of gf must satisfy:
  • x ∈ Domain of f
  • f(x) ∈ Domain of g

1.3 Function Notation

Standard Notations

• f(x) means “function f with input x”
• Examples:
  • f(x) = 2e^x
  • f: x ↦ log x for x > 0

Other Notations

• f⁻¹(x) = inverse function
• fg(x) = f(g(x)) = composite function
• f²(x) = f(f(x))

1.4 Modulus Transformations: y = |f(x)|

Understanding |f(x)|

• Modulus means all negative f(x) values become positive.
• Graph transformation:
  • Keep all f(x) ≥ 0 points unchanged.
  • Reflect all f(x) < 0 parts of the graph across the x-axis.

Examples

• f(x) = x – 2 → |f(x)| = |x – 2| results in a V-shaped graph.
• f(x) = –x² + 4 becomes positive for negative output values.

1.5 Explaining Why a Function Has No Inverse

• A function has no inverse if it is not one–one.
• Example: f(x) = x² is not one–one on all ℝ since f(2) = 4 = f(–2).
• But if you restrict the domain to x ≥ 0, then it becomes one–one.

1.6 Finding the Inverse of a One–One Function

Method to Find Inverse

1. Start with f(x), e.g., f(x) = 3x + 2
2. Write y = 3x + 2
3. Solve for x:
   x = (y – 2)/3
4. Interchange variables:
   f⁻¹(x) = (x – 2)/3

Examples

• f(x) = e^(2x) → f⁻¹(x) = (1/2)ln(x)

1.7 Form and Use Composite Functions

Order Matters

• fg(x) = f(g(x)) ≠ gf(x)
• Work from the inside out:
  • Evaluate g(x) first
  • Then substitute result into f

Example

Let f(x) = 2x + 1, g(x) = x²

• fg(x) = f(g(x)) = f(x²) = 2x² + 1
• gf(x) = g(f(x)) = g(2x + 1) = (2x + 1)²

1.8 Graphical Relationship Between Function and Inverse

• The graph of f⁻¹(x) is the reflection of f(x) in the line y = x.
• To sketch:
  • Reflect each point (a, b) of f(x) into (b, a) for f⁻¹(x)
• If a function is not one–one, its graph fails the horizontal line test, and it will not have an inverse that is a function unless restricted.

Extra Notes on Sketching and Problem Solving

Sketch Graphs

• Identify key features:
  • Domain, range, intercepts
  • Turning points for quadratics
  • Asymptotes for rational or exponential functions

Example Problem

Let f(x) = 1/(x – 2), find:

• Domain: x ≠ 2
• Range: y ≠ 0
• Inverse:
  1. y = 1/(x – 2)
  2. x = 1/(y – 2)
  3. Solve for y:
     y = 2 + 1/x
  4. So, f⁻¹(x) = 2 + 1/x

Key Concepts to Master

• Definition and logic of functions
• One–one vs many–one
• Inverses: when and how they exist
• Composite functions and order
• Function notation
• Graph transformations
• Domain/range analysis
• Linking algebraic and graphical approaches