5.1 Real-Valued Functions, Types, Domains & Ranges

Real-Valued Functions, Types, Domains & Ranges

In the Intermediate Mathematics Part II syllabus and the UET Combined Entry Test (ECAT), the theory of functions forms the conceptual gateway to differential and integral calculus. A solid grasp of function mapping, domain and range constraints, geometric symmetries, and transcendental identities is essential for answering high-speed multiple-choice questions without lengthy algebraic scratchwork.

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1. Definition of a Function and Geometric Validation

The Mathematical Mapping

Let $X$ and $Y$ be non-empty subsets of real numbers $\mathbb{R}$. A real-valued function $f$ from $X$ to $Y$, denoted as $f: X \to Y$, is a mathematical rule or relation that assigns to each element $x \in X$ exactly one element $y \in Y$.

• The set $X$ is termed the domain of the function, denoted $\text{Dom}(f)$.
• The set $Y$ is the codomain.
• The set of all actual outputs, $f(X) = \{f(x) \in Y \mid x \in X\}$, is the range of $f$, denoted $\text{Range}(f)$. Note that $\text{Range}(f) \subseteq Y$.

The Vertical Line Test

In the Cartesian coordinate plane, a curve represents a function $y = f(x)$ if and only if no vertical line intersects the curve at more than one point. If a vertical line $x = c$ intersects a graph at two or more distinct points $(c, y_1)$ and $(c, y_2)$, then a single input $c$ produces multiple outputs, violating the uniqueness requirement of a function.

• Relation Example: The circle $x^2 + y^2 = r^2$ is not a function of $x$ because vertical lines with $-r < x < r$ intersect the circle at two points: $y = \pm\sqrt{r^2 - x^2}$.
• Function Example: The upper semicircle $y = +\sqrt{r^2 - x^2}$ satisfies the vertical line test and is a valid real function with domain $[-r, r]$ and range $[0, r]$.

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2. Comprehensive Taxonomy of Real Functions

Functions are partitioned broadly into two grand families: algebraic functions and transcendental functions.

Algebraic Functions

An algebraic function is a function constructed using a finite number of algebraic operations (addition, subtraction, multiplication, division, and root extraction) on polynomials.

1. Polynomial Functions: $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0, \quad a_n \neq 0$

   • The natural domain of every polynomial is the entire real line $\mathbb{R} = (-\infty, \infty)$.
   • Degree $n = 0$: Constant function $f(x) = c$; range is $\{c\}$.
   • Degree $n = 1$: Linear function $f(x) = ax + b$ ($a \neq 0$); domain $\mathbb{R}$, range $\mathbb{R}$.
   • Degree $n = 2$: Quadratic function $f(x) = ax^2 + bx + c$; domain $\mathbb{R}$, range depends on vertex coordinates.

2. Rational Functions: $R(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials. The natural domain consists of all real numbers except the real roots of $Q(x)$: $\text{Dom}(R) = \{x \in \mathbb{R} \mid Q(x) \neq 0\}$

3. Radical (Irrational) Functions: Functions containing fractional exponents, such as $f(x) = \sqrt[n]{g(x)}$. For even index $n$ ($n = 2, 4, 6, \dots$), real outputs require $g(x) \ge 0$.

4. Explicit vs. Implicit Functions:

   • Explicit: $y$ is isolated as the dependent variable on one side: $y = f(x)$, e.g., $y = x^2 - 3x + 2$.
   • Implicit: $x$ and $y$ are mixed together in an equation: $F(x, y) = 0$, e.g., $x^3 + y^3 - 3axy = 0$.

Transcendental Functions

Functions that cannot be expressed as finite algebraic combinations of polynomials are termed transcendental.

1. Trigonometric Functions:

   • $\sin x$ and $\cos x$: Domain $\mathbb{R}$, Range $[-1, 1]$, fundamental period $2\pi$.
   • $\tan x$: Domain $\mathbb{R} \setminus \{(2k+1)\frac{\pi}{2} \mid k \in \mathbb{Z}\}$, Range $\mathbb{R}$, fundamental period $\pi$.
   • $\cot x$: Domain $\mathbb{R} \setminus \{k\pi \mid k \in \mathbb{Z}\}$, Range $\mathbb{R}$, fundamental period $\pi$.
   • $\sec x$: Domain $\mathbb{R} \setminus \{(2k+1)\frac{\pi}{2}\}$, Range $(-\infty, -1] \cup [1, \infty)$, period $2\pi$.
   • $\csc x$: Domain $\mathbb{R} \setminus \{k\pi\}$, Range $(-\infty, -1] \cup [1, \infty)$, period $2\pi$.

2. Inverse Trigonometric Functions: To define inverses, trigonometric domains are restricted to principal branches:

   • $y = \arcsin x$ (or $\sin^{-1} x$): Domain $[-1, 1]$, Range $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
   • $y = \arccos x$ (or $\cos^{-1} x$): Domain $[-1, 1]$, Range $[0, \pi]$.
   • $y = \arctan x$ (or $\tan^{-1} x$): Domain $\mathbb{R}$, Range $(-\frac{\pi}{2}, \frac{\pi}{2})$.
   • $y = \text{arccot } x$: Domain $\mathbb{R}$, Range $(0, \pi)$.
   • $y = \text{arcsec } x$: Domain $(-\infty, -1] \cup [1, \infty)$, Range $[0, \pi] \setminus \{\frac{\pi}{2}\}$.
   • $y = \text{arccsc } x$: Domain $(-\infty, -1] \cup [1, \infty)$, Range $[-\frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\}$.

3. Exponential Functions: $f(x) = a^x \quad (a > 0, a \neq 1)$

   • Domain: $(-\infty, \infty)$. Range: $(0, \infty)$.
   • If $a = e \approx 2.71828$, $f(x) = e^x$ is the natural exponential function.
   • Key behavior: $a^0 = 1$, $a^x > 0$ for all $x \in \mathbb{R}$. If $a > 1$, $f(x)$ is strictly increasing; if $0 < a < 1$, $f(x)$ is strictly decreasing.

4. Logarithmic Functions: $f(x) = \log_a x \quad (a > 0, a \neq 1, x > 0)$

   • The logarithmic function is the inverse of the exponential function: $y = \log_a x \iff a^y = x$.
   • Domain: $(0, \infty)$. Range: $(-\infty, \infty)$.
   • If $a = e$, we obtain the natural logarithm: $\ln x = \log_e x$.
   • Core properties: $\ln(xy) = \ln x + \ln y$, $\ln(x/y) = \ln x - \ln y$, $\ln(x^p) = p\ln x$, $\ln 1 = 0$, $\ln e = 1$.

5. Hyperbolic Functions: Hyperbolic functions are defined via combinations of exponential functions $e^x$ and $e^{-x}$:

   • Hyperbolic Sine: $\sinh x = \frac{e^x - e^{-x}}{2}, \quad \text{Dom} = \mathbb{R}, \quad \text{Range} = \mathbb{R}$
   • Hyperbolic Cosine: $\cosh x = \frac{e^x + e^{-x}}{2}, \quad \text{Dom} = \mathbb{R}, \quad \text{Range} = [1, \infty)$
   • Hyperbolic Tangent: $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}, \quad \text{Dom} = \mathbb{R}, \quad \text{Range} = (-1, 1)$
   • Hyperbolic Cotangent: $\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}$ ($x \neq 0$, Range $(-\infty, -1) \cup (1, \infty)$).
   • Hyperbolic Secant: $\text{sech } x = \frac{1}{\cosh x} = \frac{2}{e^x + e^{-x}}$ (Dom $\mathbb{R}$, Range $(0, 1]$).
   • Hyperbolic Cosecant: $\text{csch } x = \frac{1}{\sinh x} = \frac{2}{e^x - e^{-x}}$ ($x \neq 0$, Range $\mathbb{R} \setminus \{0\}$).

Fundamental Hyperbolic Identities

Hyperbolic identities mirror circular trigonometric formulas with critical sign variations:

$\cosh^2 x - \sinh^2 x = 1$ $1 - \tanh^2 x = \text{sech}^2 x$ $\coth^2 x - 1 = \text{csch}^2 x$ $\sinh(2x) = 2\sinh x \cosh x$ $\cosh(2x) = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 1 + 2\sinh^2 x$ $\sinh(x \pm y) = \sinh x \cosh y \pm \cosh x \sinh y$ $\cosh(x \pm y) = \cosh x \cosh y \pm \sinh x \sinh y$

ECAT Connection: In circular trigonometry, $\cos^2 x + \sin^2 x = 1$, whereas in hyperbolic trigonometry, $\cosh^2 x - \sinh^2 x = 1$. Remembering this minus sign prevents frequent calculation errors on the exam.

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3. Even, Odd, and Periodic Symmetries

Even Functions

A function $f$ is even if for all $x \in \text{Dom}(f)$: $f(-x) = f(x)$ Geometrically, the graph of an even function is symmetric with respect to the y-axis. If $(x, y)$ is on the graph, then $(-x, y)$ is also on the graph.

• Examples: $x^2, x^4, |x|, \cos x, \cosh x, \sec x$.

Odd Functions

A function $f$ is odd if for all $x \in \text{Dom}(f)$: $f(-x) = -f(x)$ Geometrically, the graph of an odd function exhibits rotational symmetry of $180^\circ$ about the origin $(0, 0)$. If $(x, y)$ is on the graph, then $(-x, -y)$ is also on the graph.

• Examples: $x, x^3, \frac{1}{x}, \sin x, \tan x, \sinh x, \csc x$.

Neither Even Nor Odd

Most functions are neither even nor odd, such as $f(x) = x^2 + 2x - 3$ or $f(x) = e^x$.

Universal Parity Decomposition Theorem

Any real-valued function $f(x)$ whose domain is symmetric about the origin can be uniquely decomposed into the sum of an even component $f_e(x)$ and an odd component $f_o(x)$:

$f(x) = f_e(x) + f_o(x)$ $f_e(x) = \frac{f(x) + f(-x)}{2} \quad (\text{even}), \qquad f_o(x) = \frac{f(x) - f(-x)}{2} \quad (\text{odd})$

Applying this to the natural exponential $e^x$ directly yields the hyperbolic definitions: $e^x = \frac{e^x + e^{-x}}{2} + \frac{e^x - e^{-x}}{2} = \cosh x + \sinh x$ where $\cosh x$ is the even component and $\sinh x$ is the odd component.

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4. Function Algebra, Composition & Inverses

Composition of Functions

Given two functions $f$ and $g$, the composite function $(f \circ g)(x)$ is defined by: $(f \circ g)(x) = f(g(x))$ The natural domain of $f \circ g$ consists of all $x$ in the domain of $g$ for which $g(x)$ belongs to the domain of $f$: $\text{Dom}(f \circ g) = \{x \in \text{Dom}(g) \mid g(x) \in \text{Dom}(f)\}$

Important: Composition is non-commutative in general: $(f \circ g)(x) \neq (g \circ f)(x)$. However, composition is associative: $f \circ (g \circ h) = (f \circ g) \circ h$.

Injectivity, Surjectivity, and Bijectivity

1. One-to-One (Injective): A function $f$ is injective if distinct inputs yield distinct outputs: $f(x_1) = f(x_2) \implies x_1 = x_2$ Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph at more than one point.
2. Onto (Surjective): A function $f: X \to Y$ is surjective if every element in the codomain $Y$ is the image of at least one element in the domain $X$, meaning $\text{Range}(f) = Y$.
3. Bijective (Invertible): A function that is both injective and surjective is bijective. A function possesses an inverse $f^{-1}$ if and only if it is bijective.

Properties of the Inverse Function $f^{-1}$

If $f$ is a bijection, its inverse $f^{-1}$ satisfies: $(f \circ f^{-1})(y) = y \quad \text{and} \quad (f^{-1} \circ f)(x) = x$

• $\text{Dom}(f^{-1}) = \text{Range}(f)$
• $\text{Range}(f^{-1}) = \text{Dom}(f)$
• Geometrically, the graph of $y = f^{-1}(x)$ is the reflection of the graph of $y = f(x)$ across the line of symmetry $y = x$.

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5. Systematic Determination of Natural Domains and Ranges

The natural domain is the largest subset of real numbers $\mathbb{R}$ for which the function formula yields real, finite outputs.

Core Rules for Natural Domain

Systematic Determination of Range

To find the range of $y = f(x)$:

1. Inversion Method: Solve the equation $y = f(x)$ explicitly for $x$ in terms of $y$: $x = g(y)$. Then determine all values of $y$ for which $g(y)$ produces valid real values within the original domain of $f$.
2. Quadratic Extrema Method: For $f(x) = ax^2 + bx + c$, the parabola vertex is at $x = -\frac{b}{2a}$ with extremum $y_v = \frac{4ac - b^2}{4a}$.
   • If $a > 0$, $\text{Range} = [y_v, \infty)$.
   • If $a < 0$, $\text{Range} = (-\infty, y_v]$.
3. Square Root Semicircle: For $f(x) = \sqrt{a^2 - x^2}$, the domain is $[-a, a]$ and the range is $[0, a]$.
4. Linear Fractional Function: For $f(x) = \frac{ax + b}{cx + d}$ with $ad - bc \neq 0$ and $c \neq 0$:
   • Natural Domain: $\mathbb{R} \setminus \{-\frac{d}{c}\}$.
   • Natural Range: $\mathbb{R} \setminus \{\frac{a}{c}\}$.

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6. Domain and Range Reference Table for Common Functions

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7. High-Speed ECAT Algebraic Shortcuts

Shortcut 1: The Self-Inverse Linear Fractional Function

For the general linear fractional function: $f(x) = \frac{ax + b}{cx + d}$ Its inverse is computed by solving $y = \frac{ax + b}{cx + d} \implies cxy + dy = ax + b \implies x(cy - a) = -dy + b \implies x = \frac{-dy + b}{cy - a}$. Swapping variables: $f^{-1}(x) = \frac{-dx + b}{cx - a}$

ECAT Instant Rule: If $a + d = 0$ (meaning $d = -a$), then: $f^{-1}(x) = \frac{-(-a)x + b}{cx - a} = \frac{ax + b}{cx + d} = f(x)$ The function is self-inverse! For example, $f(x) = \frac{3x + 4}{2x - 3}$ has $a = 3, d = -3$, so $a + d = 0$. Thus, $f^{-1}(x) = f(x)$ instantly.

Shortcut 2: Rational Inequalities for Radicals (The Wavy Curve Method)

When solving $\sqrt{\frac{x - a}{x - b}} \ge 0$:

• Find critical roots $x = a$ and $x = b$.
• Plot on the real line: the radical requires the fraction to be non-negative $\ge 0$ and $x \neq b$.
• If $a < b$, the solution set is $(-\infty, a] \cup (b, \infty)$. Note that the open parenthesis at $b$ prevents division by zero.