Introduction to Functions
      Function definition, notation and terminology
         Domain, range and codomain
         Evaluating a function
         Composition of functions (a function of a function)
         Inverse function
Function definition, notation and terminology
A function f is a relation between two sets, called the domain and the range, such that to each element x of the domain, there is assigned exactly one element f (x) of the range.
We also say that a function is an expression or a rule that associates each element of the domain with a unique element of the codomain.
In the function notation  y = f (x), x is the independent variable or argument and y is the dependent variable or a function of the variable x, where f is a rule of association. In the notation y = f (x) we call y the value of f at x.
Thus, ƒ represents all the operations which should be performed to evaluate the function at a particular value.
Domain, range and codomain
The domain is the set of values of the independent variable of a given function, i.e., the set of all first members of the ordered pairs (x, f (x)) that constitute the function.
The range is the set of values that given function takes as its argument varies through its domain. It is the image of the domain.
The codomain is the set within which the values of a function lie, as opposed to the range, which is the set of values that the function actually takes.
Therefore, the range must be a subset of, but may or may not be identical with the codomain.
We will only consider real-valued functions of a real variable.
Evaluating a function
Evaluating a function means finding  f (x) at some specific value of x. So, evaluating a function at a constant or a variable involves substituting the constant or the variable into the expression of the function and calculate its value.
Example:  Given f (x) = -x2 + 4x - 1 find,  a)  f (-1) and  b)  f (x + 2)
Solution:  a)   f (-1) = -(-1)2 + 4(-1) - 1 = -1 - 4 - 1 = -6
                b)  f (x + 2) = -(x + 2)2 + 4(x + 2) - 1 = -(x2 + 4x + 4) + 4x + 8 - 1 = -x2 + 3
 
Composition of functions (a function of a function)
Evaluation of a function at the value of another (or the same) function is called the composition of functions, denoted as  (ƒ o g)(x) = ƒ(g(x)).
Thus, the composition is the operation that forms a single function from two given functions by plugging the second function into the first for any argument.
The composition of functions is only defined if the range of the first is contained in the domain of the second function.
Examples:  Given f (x) = - x2 + 4x - 1 and  g (x) = - x + 1 find;
                  a)  f (g (x)),     b)  g (f (x)),     c)  g (g (x))   and   d)  f (g (-1))
Solutions:  a)  f (g (x)) = ƒ(- x + 1) = - (- x + 1)2 + 4(-x + 1) - 1 = - x2 - 2x + 2
                  b)  g (f (x)) = g (- x2 + 4x - 1) = - (-x2 + 4x - 1) + 1 = x2 - 4x + 2
                  c)  g (g (x)) = g (- x + 1) = - (- x + 1) + 1 = x
                  d)  f (g (-1)) = f ( - (-1) + 1) = f (2) = - 22 + 4 2 - 1 = 3
Inverse function
The inverse function, usually written f -1, is the function whose domain and the range are respectively the range and domain of a given function f, that is
f -1(x) = y  if and only if   f (y) = x .
Thus, the composition of the inverse function and the given function returns x, which is called the identity function, i.e., 
f -1(f (x)) = x    and    f (f -1(x)) = x.
The inverse of a function undoes the procedure (or function) of the given function.
A pair of inverse functions is in inverse relation.
Example:  If given  f (x) = log2 x  then f -1(x)  = 2x  since,
   
Therefore, to obtain the inverse of a function y =  f (x), exchange the variables x and y, i.e., write x = f (y) and solve for y.  Or form the composition  f (f -1(x)) = x  and solve for f -1.
Example:  Given y = f (x) = log2 determine  f -1(x).
Solution:  a)  Rewrite  y = f (x) = log2 x  to  x = log2 y  and solve for y, which gives  y = f -1(x) = 2x.
                b)  Form f (f -1(x)) = x that is,  log2 (f -1(x)) = x and solve for f -1, which gives f -1(x) = 2x.
The graphs of a pair of inverse functions are symmetrical with respect to the line  yx.
Intermediate algebra contents
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