FUnctions - Examples

Based on the sources and our conversation history, functions are described as a special type of relation where each element in the input set is mapped to exactly one element in the output set. Abstractly, a function can be thought of as a machine that produces an output for a given input.

Here are several examples of functions found in the sources:

• Functions on Numbers:

  • The Square Function: This is a frequently used example. Given an input x, it returns x². It can be written as f(x) = x². The domain and codomain are often considered the set of real numbers (R). The range is the set of non-negative real numbers, as the output of squaring any real number is always positive or zero. The graph of this function is a parabola.
  • Linear Functions: These are functions of the form f(x) = ax + b or mx + c, where a (or m) and b (or c) are real numbers and a ≠ 0. They define a straight line when graphed. A specific example given is 3.5x + 5.7. Another linear function example is f(x) = 7x + 2 and f(x) = x.
  • Quadratic Functions: These are functions of the form f(x) = ax² + bx + c, where a ≠ 0, and a, b, c are real numbers. The graph of any quadratic function is always a parabola. An example of a shifted parabola is 5x² + 3.
  • Polynomial Functions: A general polynomial function of degree n is described as f(x) = anxⁿ + an-1xⁿ⁻¹ + ... + a₀x⁰, where an ≠ 0 and n is a natural number. They consist of mathematical terms added together, involving only addition, subtraction, multiplication, and natural exponents of variables. An example given is f(x) = x³ + 5.
  • Exponential Functions: These are of the form f(x) = aˣ, where a > 0 and a ≠ 1. The natural exponential function, f(x) = eˣ, is a specific example where e > 1. Other examples include f(x) = 2ˣ and f(x) = (1/2)ˣ.
  • Logarithmic Functions: These are of the form f(x) = logₐ(x), where a > 0 and a ≠ 1. They are the inverse of exponential functions. The natural logarithmic function is f(x) = loge x = ln x, and the common logarithmic function is f(x) = log₁₀ x = log x. The domain is the set of all positive real numbers.
  • Square Root Function: The function f(x) = √x is discussed. By convention, this usually refers to the positive square root. The domain depends on the allowed codomain; if the codomain is restricted to real numbers, the domain is [0, ∞). If complex numbers are allowed as output, the domain can be all real numbers.
  • Absolute Value Function: Denoted by f(x) = |x|, this function returns x if x ≥ 0 and -x if x < 0. It is used as an example to check for injectivity (it is not one-to-one) and continuity at x=0 (it is continuous).
  • Step Functions: Examples include the Floor function, f(x) = ⌊x⌋ (greatest integer value less than or equal to x), and the Ceiling function, f(x) = ⌈x⌉ (smallest integer value greater than or equal to x).
  • Trigonometric Functions: Examples mentioned include sin x, cos x, and tan x. f(x) = sin x is also used to check for differentiability.
  • Constant Function: f(x) = c is used to illustrate differentiation.
  • Rational Function: An example of a real-valued function given is f(x) = (5x+9)/(2x).
  • Function Defined on an Interval: f(x) = 2x - 1 defined on the interval `` is used in the context of calculating area under a curve.
  • Function used in SSE: f(x) = 2x - 2 is implicitly used in a sum squared error calculation example.
  • Bounded Function Example: f(x) = 1/(x² + 1) is shown to be a bounded function with 0 ≤ f(x) ≤ 1.

• Functions on Other Sets:

  • The “Mother Of” Relation: Defined on a set of people, this relation where (m, c) is a pair if m is the mother of c, is given as an example of a relation that is also a function because every person has exactly one mother.
  • Sequences: A real sequence is defined as a function whose domain is the set of natural numbers (N) and whose codomain is the set of real numbers (R). Examples include fn = (n + 5)/(n² + 2) and an = 1 - 1/n².
  • Allocation Relation (Functional): While the teacher-course allocation mentioned initially is a general relation, if it were the case that each course was taught by exactly one teacher (and each teacher taught at least one course they were assigned), this could be a function from Courses to Teachers. (This specific functional interpretation isn’t explicitly stated in the source, but it follows the definition of a function).
  • Table with a Key: A table where a “key” (like a student roll number) uniquely identifies a “value” (like name, date of birth, etc.) is described as being like a function.

These examples demonstrate the versatility of functions in representing various types of relationships and are foundational to concepts explored later in the course, such as calculus.