The Language of Functions

The Language of Functions

Defining Functions

  • A function is a special type of relation where each element from the domain (the input) is related to exactly one element in the codomain (the output).
  • In a function, every x-value or input (also known as the independent variable) maps to a unique y-value or output (also known as the dependent variable).
  • The domain of a function contains all possible input values, while the range consists of all possible output values.

Representation of Functions

  • Functions can be represented in different ways such as equations, graphs, or tables. Each of these representations provides unique insights into the behaviour of the function.
  • The equation form allows us to see the rule that transforms the input into the output.
  • The graph gives us a visual representation of the relationship between the inputs and outputs.
  • Tables explicitly present certain input-output pairs.

Types of Functions

  • Linear functions have graphs that are straight lines. They have a constant rate of change, defined by the slope of the line.
  • Quadratic functions are characterized by an equation that can be written in the form of ax^2 + bx + c, where a ≠ 0. Their graphs are parabolas.
  • Exponential functions involve the input appearing in the exponent, e.g. f(x) = a^x. They involve consistent multiplication rather than consistent addition or subtraction.
  • Logarithmic functions are the inverse of exponential functions, and are used to solve for the variable in the exponent in exponential functions.
  • Trigonomic functions include the sine, cosine, and tangent functions that relate to the ratios of the sides in a right triangle. They are periodic functions often used to model cyclical phenomena.

Operations on Functions

  • Functions can be added, subtracted, multiplied, and divided to create new functions.
  • The composite of two functions (denoted as fog or g∘f) is the function obtained by first applying one function, then another.
  • The inverse function of a given function f is the function that reverses the “direction” of f.

Transformation of Functions

  • Functions can be transformed in various ways including translation, stretching, reflecting, and rotating to form new functions.
  • Understanding these transformations is crucial in modelling real-world situations.