CIE A Level Maths: Pure 1 - 1.3 Functions Study Notes on Functions

Language of Functions

• Definition of a Mapping:

  • A mapping is a process that takes an 'input' from one set of values and identifies an 'output' in another set of values.

• Types of Mappings:

  • Many-to-one: Multiple different 'input' values map to the same single 'output' value.
  • One-to-many: A single 'input' value maps to multiple different 'output' values.
  • Many-to-many: Multiple 'input' values can map to multiple different 'output' values.
  • One-to-one: Each unique 'input' value maps to exactly one unique 'output' value.

• Definition of a Function:

  • A function is a specific type of mapping where every 'input' value maps to a single 'output' value.
  • Consequently, only one-to-one and many-to-one mappings qualify as functions.
  • Mappings that produce multiple possible outputs for a single input (one-to-many and many-to-many) are not considered functions.

• Function Notation:

  • Functions are commonly denoted using $f(x)$, $g(x)$, etc.
  • Example: $f(x) = x^2 - 3x + 2$.
  • An alternative notation uses the mapping arrow: $f: x \mapsto x^2 - 3x + 2$.

• Sets of Numbers:

  • Functions often involve specific sets of numbers for their domains and ranges:
    • $\mathbb{N}$: Natural numbers.
    • $\mathbb{Z}$: Integers.
    • $\mathbb{Q}$: Rational numbers.
    • $\mathbb{R}$: Real numbers.
  • These sets are hierarchical; for example, $\mathbb{N}$ is a subset of $\mathbb{Z}$.
  • $\mathbb{Z}^-$ specifically refers to the set of negative integers.

• Domain:

  • The domain is the set of all values that are permitted as 'inputs' for a function.
  • A function is not fully defined until its domain has been explicitly stated.
  • Domain Restriction: Restricting the domain of a many-to-one function can transform it into a one-to-one function.

• Range:

  • The range is the set of all possible 'output' values generated by the function.
  • The specific values within the range are entirely dependent on the defined domain.

Composite Functions

• Definition of a Composite Function:

  • A composite function occurs when one function is applied immediately after another function.
  • The 'output' of the first function applied becomes the 'input' for the subsequent function.
  • This concept is often referred to as a "function-of-a-function."

• Composite Notation:

  • Composite functions can be written in several equivalent ways:
    • $fg(x)$
    • $f(g(x))$
    • $f[g(x)]$
    • $(f \circ g)(x)$
  • All of the above notations translate to "$f$ of $g(x)$."

• Working with Composite Functions:

  • The Order of Operations: The sequence in which functions are applied is critical. In the notation $fg(x)$, always start with the function closest to the variable $x$.
    1. First, apply $g$ to $x$ to obtain the result $g(x)$.
    2. Then, apply $f$ to that previous output to obtain $f(g(x))$.
  • Non-Commutative Property: Generally, $fg(x) \neq gf(x)$. The resulting function is usually different depending on the order of application, though they can be the same in specific cases.

• Special Cases and Notation:

  • Applying a function to itself, $f(f(x))$, is written as $f^2(x)$.
  • Identity Property: A function followed by its inverse returns the original input: $f f^{-1}(x) = x$.

• Examiner Tip on Existence:

  • For the composite function $fg(x)$ to exist, the range (output) of function $g$ must be contained within the domain (allowed input) of function $f$.
  • It is possible for $fg(x)$ to exist while $gf(x)$ does not exist for certain values of $x$.

Inverse Functions

• Definition of an Inverse Function:

  • An inverse function performs the opposite operation of the original function, effectively "undoing" it.
  • It is denoted by the notation $f^{-1}(x)$.

• Condition for Existence:

  • An inverse function only exists if the original function is one-to-one.

• Graphs of Inverse Functions:

  • The graph of a function and the graph of its inverse are reflections of each other across the line $y = x$.

• Domain and Range Relationship:

  • There is a reciprocal relationship between the domain and range of a function and its inverse:
    • The range of the original function $f$ becomes the domain of the inverse function $f^{-1}$.
    • The domain of the original function $f$ becomes the range of the inverse function $f^{-1}$.

• How to Calculate an Inverse Function:

  1. Set the function equal to $y$ (e.g., $y = f(x)$).
  2. Rearrange the equation to isolate $x$ and make it the subject of the formula.
  3. Rewrite the final expression using inverse function notation ($f^{-1}(x)$).
  4. Remember that the domain must be specified to fully define the inverse function.

• Inverse Identity Properties:

  • Applying a function and then its inverse (or vice versa) always returns the initial input value:
    • $f^{-1}f(x) = x$ (for all values of $x$ in the domain of $f$)
    • $ff^{-1}(x) = x$ (for all values of $x$ in the domain of $f^{-1}$)