The word function was first used by Leibniz in 1694, to describe what are today called differentiable functions. It was later used by Euler during the mid-18th century to describe an expression or formula involving variables.
A function is a special case of a relation. A relation may be defined by an algebraic equation or as a collection of ordered pairs. A function specifies a relationship between two sets of values, where the value of the function is uniquely determined by a mathematical expression containing variables or some other clearly specified rule. Consider the equation of a circle:
Solving for y in terms of x, we get:x2 + y2 = 1
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y = ±√ 1 - x2
For each value of x in the interval (-1, 1) there are two solutions for y. This
equation specifies a relation between two variables. In this equation, the variable y is called the
dependent variable; the variable x is called the independent variable. However, since
the value of y is not unique for each value of x, the equation of a circle is not a function. Now consider:
For each value of the independent variable x in the interval (-∞, +∞) there is a unique value for the dependent variable y. So this equation does have the properties of a function.y = x2 + 2x + 1
10.1. Terminology and notation
10.2. Domain, codomain and range
10.3. Graphs of functions
10.4. Classifying functions
10.5. Continuous and discontinuous functions
10.6. Solved problems