Mathematical Functions cheatsheet

General Definition of a Function

A function is a relationship between two sets,and, such that each element of setcorresponds to a unique element of set.

Formally, a function is defined as:

Common Notations

• = Function in variable
• = Derivative ofwith respect to
• = Integral ofwith respect to
• = Limit ofasapproaches
• = Absolute value of

Domain ()

The set of all possible input values

Codomain ()

The set of all output values

Image ()

The set of all output values that the function actually takes

Types of Functions

Injective (one-to-one)

Surjective (onto)

Bijective

Injective and surjective.is a one-to-one and onto function.

Function of Two Variables

Composition of Functions

whereandare functions.

Inverse Functions

The inverse function of a functionis a functionsuch that:

Existence Condition

Forto have an inverse, it must be bijective (injective and surjective).

Method to Find the Inverse

1. Write the equation
2. Swapand
3. Solve for
4. Writeas

Periodic Functions

A functionis periodic if there exists a numbersuch that:

Linear Functions

A linear function has the form:

Where:

• is the slope or rate of change.
• is the intercept or the value ofwhen.

Domain and Range

For linear functions, the domain and range are(real numbers), unless specified otherwise.

Slope ()

Describes the inclination of the line.

• If, the function is increasing
• If, the function is decreasing

Intersection with the-axis ()

The value ofwhen.

Intersection with the-axis

Occurs when, i.e.,

Quadratic Functions (Parabola)

A quadratic function has the form:

Where:

Canonical Form

Domain

Range

Depends on the sign of. If, the range is(opens upwards). If, the range is(opens downwards).

Vertex

The vertex of the parabola is atand the value ofat this point is:

Axis of Symmetry

The parabola is symmetric with respect to the line.

Roots

The solutions ofare obtained using the quadratic formula:

Discriminant

Polynomial Functions

A polynomial function is a finite sum of terms of the form. The general form of a polynomial function of degreeis:

where

• is a non-negative integer
• are real coefficients

Degree of the Polynomial

The degree is, which is the highest exponent of.

Domain

Range

Depends on the degree and coefficients of the polynomial.

Roots

Obtained by solving.

There is no general method. They can be found using the factor theorem or through numerical methods such as Newton’s method.

Derivative

The derivative of a polynomial is:

Asymptotic Behavior

The behavior of a polynomial asis dominated by the term of highest degree.

Rational Functions

whereandare polynomials.

Domain

Vertical Asymptote

Found where.

Horizontal Asymptote

Depends on the degree ofand:

• Ifthen
• Ifthen(whereandare the leading coefficients ofand)
• Ifthen there is no horizontal asymptote.

Power Functions

A power function has the form:

Where:

• is a constant.
• .

Domain

, except for values ofthat make the expression undefined.

Derivative

The derivative ofis:

Increasing/Decreasing

Depends on the sign ofand.

• If, it is increasing for
• If, it decreases for positive values of.

Exponential Functions

An exponential function has the form:

Where:

• is a constant.
• is the base of the exponential.

Domain

Range

Increasing/Decreasing

If, the function is increasing. If, the function is decreasing.

Derivative

The derivative ofis:

Asymptotes

The function has a horizontal asymptote at.

Proportionality

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and has the form:

Where:

• ,is the base of the logarithm.

Domain

Range

Natural Logarithm

If, it is called the natural logarithm and is denoted.

Derivative

Vertical Asymptote

The function has a vertical asymptote at.

Properties