Sequences and Functions Study Notes

Sequences and Series

A sequence is an ordered set of objects or numbers that follows a specific pattern infinitely or finitely, where each entry is called a term. A series represents the description of the operation that adds all items in a sequence. The Fibonacci sequence, famously illustrated by the male honeybee family tree, follows the rule where each term after the first two is the sum of the preceding two terms: $1, 1, 2, 3, 5, 8, \dots$

An Arithmetic Progression (AP) or linear sequence maintains a constant common difference $d$ between consecutive terms, calculated as $d = u_n - u_{n-1}$. The general $n\text{th}$ term of an AP with first term $a$ is given by $U_n = a + (n - 1)d$. A Geometric Progression (GP) or exponential sequence maintains a constant common ratio $r = \frac{u_n}{u_{n-1}}$, with its $n\text{th}$ term defined as $U_n = a r^{n-1}$.

Fundamentals of Relations and Functions

Historically developed by mathematicians including René Descartes, Wilhelm Leibniz, Leonhard Euler, Nicolas Bourbáki, and John Tate, functions model real-world scenarios. A relation assigns elements from a domain set $A$ to a co-domain set $B$. The set of images in the co-domain mapped from the domain forms the range. Relations are categorized into four types: one-to-one, one-to-many, many-to-one, and many-to-many.

A function is a specialized relation where each element in the domain maps to exactly one unique element in the co-domain. Thus, only one-to-one and many-to-one relations qualify as functions. Graphically, an equation defines a function if and only if it passes the vertical line test, meaning no vertical line intersects its graph at more than one point.

Injective, Surjective, Bijective, and Inverse Functions

A function $f: A \rightarrow B$ is injective (one-to-one) if $f(a_1) = f(a_2)$ implies $a_1 = a_2$, passing the horizontal line test. It is surjective (onto) if its range equals its co-domain $B$. A function that is both injective and surjective is bijective.

An inverse function $f^{-1}(x)$ reverses the mapping from range to domain such that $f^{-1}(b) = a$ when $f(a) = b$. Inverse functions exist only for bijective functions, and their graphs are reflections of the original function across the line $y = x$

Composite Functions

A composite function combines two or more functions where the output of one function becomes the input of another, represented as $(f \circ g)(x) = f(g(x))$. Composite functions generally do not commute, meaning $(f \circ g)(x) \neq (g \circ f)(x)$ unless $f(x) = g(x)$. Key identities include $(f^{-1} \circ f)(a) = a$, $(f \circ f^{-1})(b) = b$, and $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$.

Linear and Parabolic Functions

Linear functions have a constant rate of change and take the slope-intercept form $y = m x + c$, where $m = \frac{y_2 - y_1}{x_2 - x_1}$ is the gradient and $c$ is the $y$-intercept. The area enclosed by linear graphs and coordinate axes forms geometric shapes such as right triangles or trapeziums, which are calculated using standard geometric formulas.

Parabolic or quadratic functions are degree-two polynomial functions written as $y = a x^2 + b x + c$ with $a \neq 0$. If $a > 0$, the parabola opens upwards with a minimum turning point; if $a < 0$, it opens downwards with a maximum turning point. The quadratic formula derived by completing the square is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The discriminant $b^2 - 4ac$ determines root properties: real and distinct if greater than zero, real and equal if zero, and complex if less than zero. Additionally, the sum of roots is $\alpha + \beta = -\frac{b}{a}$ and the product of roots is $\alpha \beta = \frac{c}{a}$.

Polynomial Functions, Remainder, and Factor Theorems

A polynomial function of degree $n$ is expressed as $P_n(x) = a x^n + b x^{n-1} + c x^{n-2} + \dots$ where exponents are non-negative integers. Arithmetic operations follow algebraic rules where only like terms combine, and division can be performed using long division or synthetic division.

The Remainder Theorem states that dividing a polynomial $P(x)$ by a linear divisor $(x - a)$ yields a remainder $r = P(a)$. The Factor Theorem specifies that if $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$.

Rational Functions and Partial Fractions

A rational function takes the form $R(x) = \frac{f(x)}{g(x)}$ where $g(x) \neq 0$. Its domain consists of all real numbers except values where $g(x) = 0$. The zeroes of a rational function occur at values where the numerator $f(x) = 0$ and $g(x) \neq 0$

Partial fraction decomposition breaks rational expressions into simpler additive fractions based on denominator factor types: linear factors $\frac{A}{x - p}$, repeated linear factors $\frac{B}{(x - p)^2}$, and irreducible quadratic factors $\frac{A x + B}{a x^2 + b x + c}$. Improper rational fractions where the numerator degree is greater than or equal to the denominator degree require long division before decomposition.

Systems of Linear Inequalities and Linear Programming

Linear inequalities in two variables define solution half-planes bounded by lines, using dashed lines for strict inequalities and solid lines for non-strict inequalities. Linear programming maximizes or minimizes an objective function $f(x, y) = a x + b y$ over a feasible region bounded by a system of linear constraints, where optimal solutions occur at the vertices of the polygonal feasible region.