A computational graph is a directed dependency graph where nodes represent mathematical operations or variables, mapping the functional dependency from inputs to outputs. When evaluating a vector-valued function, this graph is traversed in a forward direction. Conversely, to compute gradients automatically, the graph is traced in a backwards direction from output to input, which involves multiplying matrices of partial derivatives along the paths.

Computational Graph

The partial derivative of a multivariate function $$y = f(x_1, x_2, \ldots, x_n)$$ with respect to its $$i^	extrm{th}$$ parameter $$x_i$$ measures how the function changes as $$x_i$$ increases while treating all other variables as constants. Formally, it is defined as the limit: $$\frac{\partial y}{\partial x_i} = \lim_{h ightarrow 0} \frac{f(x_1, \ldots, x_{i-1}, x_i+h, x_{i+1}, \ldots, x_n) - f(x_1, \ldots, x_i, \ldots, x_n)}{h}$$.

Partial Derivative

While single-variable calculus deals with functions of one variable, deep learning often requires working with functions of many variables, known as multivariate functions. A multivariate function takes an $$n$$-dimensional input, representing multiple parameters, and evaluates them to produce an output. This is typically written as $$y = f(x_1, x_2, \ldots, x_n)$$, where the variables $$x_1$$ through $$x_n$$ jointly determine the value of the function.

Claude

A mathematical function is a specific type of relation that assigns to each element in its domain exactly one element in its range. In terms of ordered pairs, a relation is a function if and only if each $$x$$-value (input) is matched with exactly one $$y$$-value (output).

Mathematical Function

Dive into Deep Learning

To determine whether a relation given as a set of ordered pairs is a mathematical function, examine the first coordinates. If each $$x$$-value is paired with only one $$y$$-value, the relation is a function. However, if any single $$x$$-value corresponds to two or more different $$y$$-values, the relation is not a function.

Determining if a Set of Ordered Pairs is a Function

The vertical line test is a visual method used to determine whether a set of points graphed in a rectangular coordinate system represents a function. According to this test, if any vertical line intersects the graph at more than one point, the graph does not represent a mathematical function, because this indicates a single $$x$$-value (input) is mapped to multiple $$y$$-values (outputs). Conversely, if every possible vertical line drawn on the coordinate plane intersects the graph in at most one point, then every $$x$$-value corresponds to exactly one $$y$$-value, confirming that the relation is indeed a function.

Vertical Line Test

The cube function is formally defined by the equation $$f(x) = x^3$$. Since any real numerical value can be safely substituted for $$x$$, its domain consists of all real numbers, which is written in interval notation as $$(-\infty, \infty)$$. Furthermore, cubing a positive value results in a positive output, cubing a negative value results in a negative output, and cubing zero produces zero; thus, the range also encompasses all real numbers, or $$(-\infty, \infty)$$. Its graphical representation is a distinct, smooth curve that flows continuously through the origin $$(0, 0)$$ and stretches infinitely outward in both directions.

Cube Function

The square root function is algebraically defined by the equation $$f(x) = \sqrt{x}$$. Since the square root of a negative value does not produce a real number, the permissible inputs are strictly limited to non-negative real numbers, making the domain $$[0, \infty)$$. Additionally, because the principal square root is defined to be non-negative, the resulting outputs are also zero or positive, establishing the range as $$[0, \infty)$$. Visually, its graph is a steady curve that originates exactly at the origin $$(0, 0)$$ and arcs upward and to the right.

Square Root Function

A **polynomial function** is a function whose range values are defined by a polynomial expression. For example, the functions $$f(x) = x^2 + 5x + 6$$ and $$g(x) = 3x - 4$$ are considered polynomial functions because their defining expressions, $$x^2 + 5x + 6$$ and $$3x - 4$$, are polynomials.

Polynomial Function

For any mathematical function $$f$$, if $$f(x) = 0$$, then $$x$$ is a zero of the function. Because the function's output is zero at this input value, the corresponding point $$(x, 0)$$ represents an x-intercept on the graph of the function.

Zero of a Function

A limit describes the behavior of a mathematical expression as a specified variable approaches a particular value. In calculus, a common application is finding the value that a ratio, such as the change in function value over a perturbation size $$\frac{f(x+h) - f(x)}{h}$$, converges to as the perturbation $$h$$ shrinks to zero.

Limit of a Function

Multivariate Function

In mathematics, equations that define functions are often written using function notation. Functions are typically assigned single-letter names, most commonly $$f$$, $$g$$, $$h$$, $$F$$, $$G$$, or $$H$$. For a function named $$f$$, the notation specifies the relationship between an input value $$x$$ and its corresponding output value $$y$$ as $$y = f(x)$$. In this notation, $$x$$ represents the domain value, and $$f(x)$$ represents the corresponding range value. The expression $$f(x)$$ is read as '$$f$$ of $$x$$' or 'the value of $$f$$ at $$x$$'. The parentheses in this context indicate the input parameter mapping to the function and do not indicate multiplication.

Function Notation

A function is one-to-one if each value in the range corresponds to exactly one element in the domain. Equivalently, for each ordered pair in the function, every $$y$$-value is paired with only one $$x$$-value, meaning there are no repeated $$y$$-values among different ordered pairs. While a standard function requires that each $$x$$-value maps to only one $$y$$-value, a one-to-one function adds the additional requirement that each $$y$$-value maps back to only one $$x$$-value.

One-to-One Function

A relatable way to understand mathematical functions is through the analogy of birthdays. A relation mapping people to their birthdays represents a mathematical function because every person in the domain has exactly one birthday in the range. It is perfectly acceptable for two different people to share the same birthday, just as two different $$x$$-values can correspond to the same $$y$$-value in a function. However, if one person were to have two different birthdays, it would violate the definition of a mathematical function, illustrating how a single $$x$$-value cannot map to multiple $$y$$-values. Notably, this birthday relation is not a one-to-one function because two different people (such as Liz and June) can share the same birthday (August 2). Since one range value maps to two domain values, the function fails the one-to-one requirement even though it is still a valid function.

Birthday Analogy for Functions

An exponential function is formally defined as a mathematical function of the form $$f(x) = a^x$$, where the base $$a$$ is a constant such that $$a > 0$$ and $$a 
eq 1$$. Unlike linear or quadratic functions where the variable is the base, in an exponential function, the variable $$x$$ is the exponent. Exponential functions are essential for modeling many naturally occurring phenomena, including population growth, the spread of viruses, radioactive decay, and compounding interest.

Exponential Function

A sequence is a function whose domain is restricted to the counting numbers ($$1, 2, 3, \dots$$). Rather than mapping any real-number input to an output, a sequence pairs each counting number with exactly one value. The resulting outputs, called terms, are conventionally listed in order: for example, $$2, 4, 6, 8, 10, \dots$$ is a sequence. Each number in this ordered list occupies a specific position determined by its corresponding counting number. A sequence can also be viewed simply as an ordered list of numbers where position matters.

Sequence (Algebra)

In a professional data management system, a rule that assigns values from a set of inputs (the domain) to a set of outputs (the range) is classified as a **mathematical function** if and only if which of the following conditions is met?

In a professional database, a rule that maps each unique employee ID number to an office location is classified as a **mathematical function** if each employee ID is associated with exactly one office location.

In a professional data management system, maintaining consistent records is vital. Match each term below with its correct mathematical definition to ensure the mapping of data follows the rules of a function.

In a corporate payroll database, a rule maps each unique 'Employee ID' to a 'Base Salary'. For this mapping to be classified as a **mathematical function**, what is the specific requirement regarding the number of base salaries that can be assigned to any single Employee ID?

Data Integrity and Functions

In a corporate data architecture, a mapping is defined as a **mathematical function** if it assigns to each $x$-value (input) in its domain exactly ____ $y$-value (output) in its range.

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