# 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 *x _{0}* (I

_{x0}) is any interval containing

*x*. If

_{0}*x*is the midpoint of the interval, it is called a circular neighborhood with radius:

_{0}*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 *x _{0}* 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 *x _{0}* or infinity, then there exists a neighborhood of

*x*(or a sufficiently large interval for infinity) where the function has the same sign as the limit.

_{0}*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, *x _{0}* is an accumulation point, and:

=
= *L*

and there exists a neighborhood of *x _{0}* 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 *x _{0}* in (

*a*,

*b*) such that

*f(x*.

_{0}) = 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'(x _{0}) = 0*), it’s called a horizontal inflection point. If the second derivative changes sign at

*x*, it’s an inflection point with an angled tangent. If the first derivative approaches infinity at

_{0}*x*and the second derivative changes sign, it’s a vertical inflection point.

_{0}