How to Understand Functions

In mathematics, a function is a relation between a set of inputs (often referred to as the domain) and a set of possible outputs (often called the range). Each input is related to exactly one output. This means that for every element in the domain, there is a unique element in the range that the function maps to.

How to Understand Functions

Step-by-step Guide to Understand Functions

Here is a step-by-step guide to understand functions:

Step 1: Grasp the Fundamental Definition

  • A function is a special type of relation where each input (from the domain) corresponds to exactly one output (in the range).

Step 2: Differentiate Between Relations and Functions

  • Not all relations are functions. A vertical line test can determine this visually: if a vertical line intersects a graph at more than one point, it’s not a function.

Step 3: Explore Various Representations

  • Functions can be presented in multiple formats: algebraic equations, graphs, tables, or even verbal descriptions. Familiarize yourself with these variations.

Step 4: Delve into Function Notation

  • Understand the notation \(f(x)\) where \(f\) denotes the function, and \(x\) is the input. The entire expression represents the output for the given \(x\).

Step 5: Understand the Domain and Range in-depth

  • As discussed previously, the domain comprises possible input values, while the range consists of resultant output values. For complex functions, determining these might involve solving inequalities or analyzing asymptotic behavior.

Step 6: Explore Different Types of Functions

  • Linear Functions: Represented by straight lines.
  • Quadratic Functions: Parabolic in nature.
  • Exponential and Logarithmic Functions: Deal with growth and decay.
  • Trigonometric Functions: Originating from circle geometry.
  • Rational, Radical, and Polynomial Functions: Various algebraic forms with their peculiarities.
  • Piecewise Functions: Defined in pieces, each valid in a specific interval.
  • Implicit Functions: Not explicitly solved for one variable.

Step 7: Dive into Composite and Inverse Functions

  • Composite Functions: Formed by applying one function after another \((f(g(x)))\).
  • Inverse Functions: Switch the roles of input and output. If \(y=f(x)\), then the inverse is denoted as \(f\)−1\((y)\).

Step 8: Examine Function Transformations

  • Familiarize yourself with how functions change with transformations: translations (shifts), dilations (stretches/shrinks), and reflections.

Step 9: Study Limit Behavior and Continuity

  • Understand the concept of limits, and how functions behave as they approach certain values.
  • Explore what it means for a function to be continuous or discontinuous at a point.

Step 10: Operate with Functions

  • Learn to add, subtract, multiply, and divide functions.
  • Understand the resultant domain restrictions.

Step 11: Experiment with Real-world Applications

  • Recognize how functions model various real-world phenomena, from population growth to sound waves.

Step 12: Engage in Advanced Topics (for the avid learner)

  • Taylor and Maclaurin Series: Approximating functions with polynomials.
  • Fourier Series: Decomposing functions into sinusoidal components.
  • Laplace and Z-transforms: Used in engineering and control systems.

Step 13: Continual Reflection and Exploration

  • Revisit concepts, practice with various problems, and seek deeper understanding.
  • Use tools like graphing calculators or software (e.g., Desmos, MATLAB) to visualize and experiment.

Understanding functions, especially amidst high variation and complexity, is a journey. It’s an interplay between theory, application, and intuition. Embrace the intricacies, and remember that mastery comes with patience and practice.


Example 1:

Given the set of ordered pairs \(S=\){\((4,6),(5,7),(6,8),(4,9)\)}, does \(S\) represent a function?


Notice that the input value \(4\) corresponds to both \(6\) and \(9\). Since the same input has two different outputs, \(S\) does not represent a function.

Example 2:

Given the set of ordered pairs \(T=\){\((−2,0),(0,−1),(2,1),(3,2)\)}, does \(T\) represent a function?


Every input value in \(T\) corresponds to exactly one output. Therefore, \(T\) does represent a function.

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