Real Functions of Real Variables - Comprehensive Notes

Definition and Basic Concepts

• Function definition: A function is a rule that assigns to every element of its domain exactly one element of its codomain.
  • Notation: if a function has domain A and codomain B, we write $f: A \to B$ and require that for every $x \in A$ there exists a unique $y \in B$ with $y = f(x)$, i.e. $\forall x \in A, \exists! y \in B : y = f(x).$
• Domain, codomain, and range
  • Domain: the set of inputs for which the function is defined (often denoted A).
  • Codomain: the set in which the function values are allowed to lie (often denoted B).
  • Range (image): the actual values attained by the function, denoted \(F(A) = { y \in B : \exists x \in A \text{ with } y = f(x) } \
  • Note: the range is always a subset of the codomain: \
    $F(A) \subseteq B.$
• Graph of a function
  • Graph(f) = { (x, y) : x \in A, y = f(x) }\, viewed in the Cartesian plane.
  • For real-valued functions with domain A ⊆ (\mathbb{R}) and codomain B ⊆ (\mathbb{R}), the graph lies in the plane and encodes the same information as the formula.
• Natural domain and restricted domains
  • Natural domain: the largest set on which the formula defining the function yields valid outputs for all inputs.
  • If one restricts the domain to a smaller set than the natural domain, this is a deliberate restriction and should be stated explicitly.

Examples of Functions and Notation

• Linear function example
  • If $f(x) = mx + c$ with domain $A = \mathbb{R}$ and codomain $B = \mathbb{R}$:
  • If $m \neq 0$, the range is all real numbers: $f(\mathbb{R}) = \mathbb{R}.$
  • If $m = 0$, then $f(x) = c$ is constant and the range is ${c}$.
• Polynomial example
  • $f(x) = ax^2 + bx + c$: a parabola
  • Axis of symmetry: $x = -\dfrac{b}{2a}$ (for $a \neq 0$)
  • Range depends on the sign of $a$ and the vertex value; e.g., if a > 0, the minimum occurs at the vertex.
• Common non-linear examples
  • Inverse proportionality / rectangular hyperbola: $f(x) = \dfrac{1}{x}$
  • Domain: $\mathbb{R} \setminus {0}$
  • Range: $\mathbb{R} \setminus {0}$
  • Square-root function: $f(x) = \sqrt{x}$
  • Domain: $[0, \infty)$
  • Range: $[0, \infty)$
• Square and exponents
  • Example: $f(x) = \sqrt{x}$ with domain $[0, \infty)$ has inverse $f^{-1}(x) = x^2$ on the domain $[0, \infty)$.
  • Example: $f(x) = x^2$ on $\mathbb{R}$ is not injective and hence not invertible on the whole real line; restricting the domain to $[0, \infty)$ makes it invertible with inverse $f^{-1}(x) = \sqrt{x}$.
• Inverse function and invertibility
  • An inverse function exists if and only if the function is invertible (bijective) on its domain.
  • If $y = f(x)$ and the inverse exists, then
  • $f^{-1}(f(x)) = x\quad(\text{for } x\in A)$
  • $f(f^{-1}(y)) = y\quad(\text{for } y \in F(A)).$
  • The graph of a function and the graph of its inverse are symmetric about the line $y = x$.
• Inverse notation and key ideas
  • The inverse function is often denoted $f^{-1}$, with domain $F(A)$ and codomain $A$:
  • $f^{-1}: F(A) \to A,\quad f^{-1}(y) = x \text{ where } y = f(x).$
  • A compact way to express the inverse relation is: if $y = f(x)$, then $x = f^{-1}(y)$.

Injective, Surjective, and Bijective Functions

• Injective (one-to-one)
  • Definition: for all $x<em>1, x</em>2 \in A$, if $f(x<em>1) = f(x</em>2)$ then $x<em>1 = x</em>2$.
  • Graphical test: horizontal line test. A horizontal line y = c intersects the graph at most once for all c.
• Surjective (onto)
  • Definition: ∀ y ∈ B, there exists at least one $x ∈ A$ with $f(x) = y$.
  • Equivalently, the range of f equals the codomain: $F(A) = B.$
• Bijective
  • Definition: both injective and surjective.
• Examples and notes
  • The earlier examples show functions that are injective but not surjective, surjective but not injective, and both (bijective).
• Important takeaway
  • A function is invertible if and only if it is bijective; equivalently, the inverse exists on the range of the function and maps back to the domain.

Graphical Criteria and Tests

• Vertical line test (function test)
  • A graph represents a function if every vertical line intersects the graph at most once.
• Horizontal line test (injectivity)
  • A function is injective if every horizontal line intersects the graph at most once.
• Invertibility and the horizontal line test
  • Invertibility requires injectivity; the horizontal line test can be used to judge whether a function is invertible on its domain.
• Example checks
  • Example: (f(x) = x^3) is invertible with inverse (f^{-1}(x) = \sqrt[3]{x}).
  • Example: (f(x) = x^2) on (\mathbb{R}) is not invertible (not injective), but on ([0, \infty)) it is invertible with inverse (f^{-1}(x) = \sqrt{x}).
• Graphical interpretation of inverse
  • If the graph of a function and the graph of its inverse are drawn, they are mirror images across the line $y = x$.

Composition of Functions

• Definition and order of application
  • If $f: A \to B$ and $g: B \to C$, the composition is $h = g \circ f: A \to C$ defined by
    $h(x) = g(f(x)).$
  • The order is read from inside to outside: apply $f$ first, then $g$.
• Example
  • Let $f(x) = x^2$ and $g(x) = 3x + 2$.
  • Then
    $(g \circ f)(x) = g(f(x)) = g(x^2) = 3x^2 + 2.$
  • And
    $(f \circ g)(x) = f(g(x)) = (3x + 2)^2.$
  • In general, composition is not commutative: $g \circ f \neq f \circ g$ in most cases.
• Practical notes
  • When writing compositions, read from inside to outside; express the inner function first, then the outer function.

The Inverse Function and Its Properties

• Definition recap
  • A function is invertible if there exists an inverse function such that
    $f^{-1}: F(A) \to A,\quad f^{-1}(y) = x \text{ where } y = f(x).$
• Existence conditions
  • A sufficient and standard condition: if $f: A \to B$ is bijective, then the inverse $f^{-1}: B \to A$ exists uniquely.
  • If the codomain is chosen as the range, i.e., if we take $f: A \to F(A)$, then injectivity guarantees invertibility on the range.
• Examples
  • $f(x) = x^3$: invertible on (\mathbb{R}) with inverse $f^{-1}(x) = \sqrt[3]{x}$.
  • $f(x) = \sqrt{x}$ with domain ([0, \infty)$: inverse is$f^{-1}(x) = x^2$with domain ([0, \infty)$.
  • $f(x) = x^2$ on (\mathbb{R}): not invertible; restricting domain to ([0, \infty)) yields inverse $f^{-1}(x) = \sqrt{x}$.
• Notation and conventions
  • It is common to write the inverse as $f^{-1}$; some authors use alternative notations in specific contexts, but the standard is as above.
• Graphical relationship
  • Graphs of a function and its inverse are symmetric with respect to the line $y = x$.
• How to find an inverse in practice
  • Solve the equation $y = f(x)$ for x in terms of y, then swap x and y to express the inverse: $y = f(x) \Rightarrow x = f^{-1}(y).$
• Important caveats
  • Inverse functions must be defined on the domain that corresponds to the range of the original function; if the original function has a restricted range, its inverse is defined on that restricted range.

Summary of Key Concepts to Remember

• A function f: A → B assigns exactly one value in B to each element of A: $\forall x ∈ A, \exists! y ∈ B : y = f(x).$
• Image vs preimage
  • Image (range): F(A) = { y ∈ B : ∃ x ∈ A, y = f(x) }.$n- Preimage: for a subset S ⊆ B, f^{-1}(S) = { x ∈ A : f(x) ∈ S }.$For a single point y ∈ B,$f^{-1}({y}) = { x ∈ A : f(x) = y }.$$
• Injective, Surjective, Bijective
  • Injective: distinct inputs map to distinct outputs.
  • Surjective: all codomain elements are attained.
  • Bijective: both properties hold.
• Invertibility and inverse functions
  • Invertible iff bijective on the given domains; inverse function exists on the range of f.
  • Graphs mirror across the line y = x.
• Graphical tests
  • Vertical line test ensures the function property; Horizontal line test ensures injectivity and, hence, invertibility (when combined with the range considerations).
• Composition of functions
  • (g ∘ f)(x) = g(f(x)); order of application matters; not generally commutative.
• Examples to solidify ideas
  • Linear: f(x) = mx + c; may be bijective if m ≠ 0 and domain/codomain are appropriate.
  • Polynomial: f(x) = ax^2 + bx + c; invertible only after restricting domain.
  • Rational: f(x) = 1/x, domain ℝ \ {0}, range ℝ \ {0}.
  • Root: f(x) = √x, domain [0, ∞), range [0, ∞); inverse x^2 with domain [0, ∞).
  • Cube: f(x) = x^3, invertible with inverse ∛x.

Examples for Quick Reference

• Invertible functions on restricted domains
  • f(x) = x^2 on [0, ∞) → [0, ∞), inverse f^{-1}(x) = √x.
  • f(x) = √x on [0, ∞) → [0, ∞), inverse f^{-1}(x) = x^2.
• Common domains and ranges
  • f(x) = mx + c, m ≠ 0: domain = all real numbers; range = all real numbers.
  • f(x) = x^2: domain = ℝ, range = [0, ∞) if codomain is ℝ; not surjective if codomain is ℝ.
  • f(x) = 1/x: domain = ℝ \ {0}, range = ℝ \ {0}.
  • f(x) = √x: domain = [0, ∞), range = [0, ∞).
• Graphical interpretation
  • The graph of f and f^{-1} are reflections across the line y = x.
• Useful conventions
  • When writing inverse functions, keep track of domains and codomains to ensure the inverse is well-defined.
• Theoretical note
  • The existence of an inverse is intimately tied to injectivity and the mapping between the domain and the range; the inverse function serves to “undo” the original mapping on its range.