Understanding Mathematical Functions and Their Properties
Functions
A function, as defined by Laplace, is a law that associates each value of the independent variable x with a value of the dependent variable y. This correspondence can be written as:
A function is said to be injective if each element of the domain (set A, represented by x) maps to a unique element of the codomain (set B, represented by y). This is also known as a one-to-one correspondence.
A function is surjective if every element of the codomain (B) is an image of an element in the domain (A). If not all elements of B are images of A, the function is not surjective.
A function that is both injective and surjective is called bijective.
The domain of a function is the set of all real values (from -∞ to +∞) of the independent variable (x) for which the function is defined.
The codomain of a function is the set of all possible values of the dependent variable (y).
Algebraic functions are obtained by repeated application of basic arithmetic operations (addition, subtraction, multiplication, division) and raising to a power. The relationship between x and y, in implicit form, can be expressed as an algebraic equation.
Transcendental functions are those that cannot be expressed solely through algebraic operations. Examples include:
A parabola is not an injective function.
A full circle is not a function. A semicircle can be an injective function.
A hyperbola is not a function. A branch of a hyperbola can be an injective function.
Sine and cosine functions are not injective (because multiple x values can have the same y value).
The logarithm is an injective function because it is either strictly increasing or strictly decreasing.
Intervals and Neighborhoods
An interval between a and b is the set of all numbers between a and b. If the interval is closed, a and b are called the minimum and maximum of the interval, respectively. If the interval is open, they are called the lower and upper bounds.
There are two types of neighborhoods: limited and unlimited.
A limited neighborhood around x0 (Ix0) is any interval containing x0. If x0 is the midpoint of the interval, it is called a circular neighborhood with radius:
Unlimited neighborhoods include all numbers from a point to infinity:
or all numbers from negative infinity to a point:
Calculating the Domain of a Function
The domain of a function is represented by the set of real numbers for which the function is defined.
For a polynomial function (e.g., a sum of terms like):
the domain is all real numbers.
For a rational function (a ratio of polynomials), the denominator restricts the domain (x values that make the denominator zero are excluded). Set the denominator equal to 0 to find these values:
Irrational Functions
Even numbers:
Odd numbers:
A logarithmic function is defined only for positive arguments.
Restrictions on exponential functions relate to the exponent.
Uniqueness of the Limit Theorem
If a function, as x approaches x0 or infinity, admits a limit, then this limit is unique.
Permanence of Sign Theorem
If a function approaches a limit l > 0 or l < 0 as x approaches x0 or infinity, then there exists a neighborhood of x0 (or a sufficiently large interval for infinity) where the function has the same sign as the limit.
- l > 0 implies f is positive
- l < 0 implies f is negative
Comparison Theorem
This theorem helps calculate the limit of a function by comparing it with two other functions. If f(x), g(x), and h(x) are defined on a domain, x0 is an accumulation point, and:
=
= L
and there exists a neighborhood of x0 such that f(x) ≤ g(x) ≤ h(x), then:
= L
Weierstrass Theorem
Every real-valued function that is continuous on a closed and bounded interval has an absolute maximum and an absolute minimum.
A corollary of the Weierstrass theorem states that a continuous function on a closed and bounded interval assumes all values between its minimum and maximum.
Existence Theorem of Zeros
If a continuous function y = f(x) on a closed interval has a negative minimum and a positive maximum, then there exists at least one point c in the interval such that f(c) = 0.
Mathematically:
Let f be continuous on [a, b] such that f(a) * f(b) < 0. Then there exists at least one x0 in (a, b) such that f(x0) = 0.
Asymptotes
An asymptote is a line that a curve approaches but never touches.
The Second Derivative
The second derivative is the derivative of the first derivative. Its sign determines the concavity of a function. The curve is concave up where the second derivative is positive and concave down where it is negative. Points where the concavity changes are called inflection points.
Rolle’s Theorem
If a function f(x) is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0.
Rolle’s theorem is a special case of Lagrange’s theorem.
Lagrange’s Theorem (Mean Value Theorem)
If a function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that:
f'(c) = (f(b) – f(a)) / (b – a)
Inflection Points of a Function
An inflection point is a point where the curve changes concavity. If the tangent at the inflection point is horizontal (f'(x0) = 0), it’s called a horizontal inflection point. If the second derivative changes sign at x0, it’s an inflection point with an angled tangent. If the first derivative approaches infinity at x0 and the second derivative changes sign, it’s a vertical inflection point.