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Function Fundamentals

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Definition. A function f is a rule, relationship, or correspondence that assigns to each element of one set (x ∈ D), called the domain, exactly one element of a second set, called the range (y ∈ E).

The pair (x, y) is denoted as y = f(x). Typically, the sets D and E will be both the set of real numbers, ℝ.

Let f be a function, a, b ∈ Domain(f). If a = b, then f(a) = f(b), e.g., f: ℚ → ℚ, f(p/q) = -p is not a function because f(12)=-1≠-2=f(24), but 12 = 24.

The domain of a function is the set of all inputs, in other words, it consists of all the elements for which f is defined on. The range of a function is the set of all outputs.


There are many was to represent a function:

If a domain is not stated explicitly, the domain is the set of all numbers for which the expression makes sense and defines a real number, e.g., f(x) = x3 + 2x2 + 4, Domain(f) = ℝ; g(x) = $\sqrt{x-2}$, Domain(g) = ℝ - {x ≥ 2} = [2, +∞); h(x) = $\frac{2x+3}{x^2-x}+4,$ Domain(h) = (-∞, 0) ∪ (0, 1) ∪ (1, ∞). r(x) = $\frac{\sqrt{x}}{x^2 - x},$ Domain(r) = (0, 1) ∪ (1, ∞) = {x ∈ ℝ: 0 < x < 1 or x > 1}.


$f(x) = \begin{cases} x, &x ≥ 0 \\ -x, &x < 0 \end{cases}$

This is the absolute value or modulus module. The absolute |x| of a real number x is the non-negative value of x without regard to its sign. It is the distance of the number from zero on a number line.

$f(x) = \begin{cases} x + 1, &x > 0 \\ -2x + 2, &x < 0 \end{cases}$

The Fibonacci numbers are the numbers in the following sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, ···. By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. In mathematical terms, F(n)= F(n-1) + F(n-2) where F(0) = 1 and F(1) = 1 (Case base).

The factorial function of a whole number, written as n!, is found by multiplying n by all the whole numbers less than it, e.g., 5! = 5·4·3·2·1 = 120, 4! = 4·3·2·1 = 24. In can also be defined recursively as n! = n·(n-1)!.


Another example is x = t3 + 1, y = t2 −1, −2 ≤ t ≤ 2 (Figure 2.b.). We can also eliminate the parameter t, y = t2 −1 =[t = (x-1)1/3] = $(x-1)^{2/3}-1.$

Evaluation of a function

Evaluating a function means finding the value of f(x) that corresponds to a given value of x. Examples:

  1. Domain(f) = {1, 2, 3, 4, 5, 6, 7}. Range(f) = {2, 4, 6, 8, 10, 12, 14}
  2. f(5) = 10, f(2) = 4, f(3) = 6, but f(9) is undefined.
  3. Solve the equation f(x) = 12. The solution is x = 6. Solve the equation f(x) = 5. The solution does not exist.
  4. Solve the inequality f(x) > 10. The solution is x = 6 or 7.


The Vertical Line Test

The vertical line test is a graphical method of determining whether a curve represents the graph of a function or not by visually examining the number of intersections of the curve with vertical lines. A vertical line needs to intersect the graph of a function at most at one point for it to represent a function.

Example: f(x) = x + 2, y = x3 are functions, but y2 = x ↭ y = $±\sqrt{x}$ is not a function (e.g., x = 4, y = ±2) -Figure 1.c-. The circle x2 + y2 = 9 is not a circle either.



This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
  1. NPTEL-NOC IITM, Introduction to Galois Theory.
  2. Algebra, Second Edition, by Michael Artin.
  3. LibreTexts, Calculus. Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
  4. Field and Galois Theory, by Patrick Morandi. Springer.
  5. Michael Penn, and MathMajor.
  6. Contemporary Abstract Algebra, Joseph, A. Gallian.
  7. YouTube’s Andrew Misseldine: Calculus. College Algebra and Abstract Algebra.
  8. MIT OpenCourseWare 18.01 Single Variable Calculus, Fall 2007 and 18.02 Multivariable Calculus, Fall 2007.
  9. Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences.
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