The Definition of a Function

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The term function was coined by mathematician Gottfried Leibniz in 1673 although Oresme came close to a modern formulation of the concept of a function in the 1300s. The notation "f(x)" was introduced my Clairaut and Euler in 1734. Many famous mathematicians have devoted time to the study of functions including Bernoulli, Fourier and Weierstraß.

Introduction to Functions

A function is defined as a mapping with a single output value. This includes both one to one mappings and many to one mappings. Therefore a one to many mapping is not a function.

Domain : the set of values used as 'input' to the function

Range : the set of values of 'output' from the function.

Two functions f and g are equal if:

  • The domain of f is equal to the domain of g.
  • The range of f is equal to the range of g.
  • f(x) = g(x) for all x ε domain of f = domain of g.

Only if all three points are satisfied then we can say the two functions are equal.

Notation of Functions

You should be familiar with the notation that we use in mathematics. When describing the domain and range of a function, we usually use the following notation.

Notation Meaning
\[R^+\] \[ x\in R : x > 0 \]
\[R^+_0\] \[ x\in R : x \geq 0 \]
\[R^-\] \[ x\in R : x < 0 \]
\[R^-_0\] \[ x\in R : x \leq 0 \]


Question: Are the functions \(f: X -> Y\) and \(g: U -> V\), where \(X = Y = U = R\) and \(V = R+0\), \(f(x) = x^2\) for all \(x \in X\), \(g(u) = u^2\) for all \(u /in U\) equal?

Answer: No, because the input values \((x, u)\) can take any value, the product from both functions are positive and equal to \(0\) at \(x = 0\), so the functions are the same, the only problem is that the range of the products are different, i.e. \(R\) and \(R^+_0\).

Cartesian Product of Two Sets

Let \(X\), \(Y\) be two sets. By the Cartesian product of \(X\) and \(Y\), denoted by \(X \times Y\), is meant the set of all ordered pairs \((x,y)\) where \(x \in X\) and \(y \in Y\). Thus we find \[ X \times Y = \{(x,y): x \in X \quad \textrm{and} \quad y \in Y\} \] Let us consider a simple example.


Question: Write out in full all elements of the Cartesian product \(X \times Y\) if \(X = \{x1, x2\}\) and \(Y = \{y1,y2,y3\}\).

Answer: If \(X = \{x1, x2\}\) and \(Y = \{y1,y2,y3\}\), then \(X \times Y = \{(x1, y1), (x1, y2), (x1, y3), (x2, y1), (x2, y2), (x2, y3)\}\)

What is meant by the graph of a function \(f: X -> Y\) is meant the set of ordered pairs \((x,y)\) where \(x \in \) and \(y = f(x)\). The graph of \(f: X -> Y\) is therefore a subset of the Cartesian product \(X \times Y\). The graph of a real valued function \(f: X -> Y\) can be obtained by plotting the pairs \((x,y)\). The independent variable \(x\) is plotted on the horizontal axis, and \(y\), the dependent variable is plotted on the vertical axis.

Back to Functions.