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
     The graph of a function
      Functions behavior, properties and characteristic points of the graph
         Domain and range
         Roots or zero function values, x-intercepts, y-intercepts
         Increasing/decreasing intervals
         The instantaneous rate of change or the derivative
         Continuity and discontinuity
         Vertical, horizontal and oblique or slant asymptotes
         Stationary points and/or critical points
         Turning points (extremes, local or relative maximums or minimums)
         Inflection points and intervals of concavity
         Symmetry of a function, parity - odd and even functions
      Transformations of original or source function
         How some changes of a function notation affect the graph of the function
         Translations of the graph of a function
         Reflections of the graph of a function
Introduction to Functions
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) = f (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)) = f (- 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 form  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.
The graph of a function
The graph of a function ƒ is drawing on the Cartesian plane, plotted with respect to coordinate axes, showing functional relationship between given variables containing all those points (x, f (x)) which satisfy the given relation.
The points lying on the curve satisfy the relation that forms the shape of the graph.
The graphic representation of a function provides insight into the behavior of the function.
Functions behavior, properties and characteristic points of the graph
To sketch the graph of a function we should know its properties and find out its characteristic points, as 
  - domain and range
  - x-intercepts or zeros (roots) and the y-intercept
  - intervals of increasing and decreasing
  - continuity and discontinuity
  - vertical, horizontal and oblique or slant asymptotes
  - turning points (extremes, local or relative maximums or minimums)
  - inflection points and intervals of concavity
  - symmetry (odd and even functions) with respect to the x-axis, y-axis, and the origin
Domain and range
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.
Roots or zero function values, x-intercepts, y-intercepts
A zero of a function is a value of the argument of a function at which the value of the function is zero.
An intercept is the point at which a given function intersects with specified coordinate axis, or the value of that coordinate at that point.
An x-intercept is the point (x, 0) where the graph of the function touches or crosses the x-axis. 
That is, at the x-intercept, the coordinate y = 0.
A zero of a function is the x value of the x-intercept. The zeros (roots) of a function correspond to the x-intercepts of the graph.
The y-intercept is the value of y where the graph crosses the y-axis.
The y-intercept correspond to the point (0, y) on the y-axis therefore, at the y-intercept the coordinate x = 0.
Increasing/decreasing intervals
A function ƒ is increasing on an interval if 
f (x1) <  f (x2)  for each x1 < x in the interval.
A function ƒ is decreasing on an interval if 
f (x1) > f (x2)  for each x1 < x in the interval.
By looking at the graph of a function being traced out as the value of the input variable x increases from left to right then, if at the same time the output value y f (x) also increases, we say the function is increasing.
If the output value decreases as x increases, then we say the function ƒ is decreasing.
Thus, if the slope or gradient m of the secant line passing through the points (x1f (x1)) and (x2, f (x2)) of the graph of a function is positive, the function is increasing (going up), as shows the figure below. 
where,   x2 - x1> 0
 
 
Since the difference x2 - x1 is always positive, when the function is decreasing (going down), the slope will be negative.
The instantaneous rate of change or the derivative
The ratio of the rise and the run, called the difference quotient, that equals the value of the tangent of the angle between the direction of the secant line and x-axis, becomes the slope (gradient) of the tangent line as the difference Dx tends to zero, and is called the instantaneous rate of change or the derivative at the point of the function.
For a given function ƒ and point (x1, ƒ(x1)), the derivative of ƒ at x = x1 is the slope of the tangent line through the point (x1, ƒ(x1)), i.e.,  f '(x1) = tan at .
The gradient of a curve at a point on its graph, expressed as the slope of the tangent line at that point, represents the rate of change of the value of the function and is called derivative of the function at the point, written
y' = dy/dx =  f '(x)
Continuity and discontinuity
A function that has no sudden changes in value as the variable increases or decreases smoothly is called continuous function.
Or more formally, a real function  y = f (x) is continuous at a point a if the limit of  f (x) as x approaches a is f (a).
If a function does not satisfy this condition at a point it is said to be discontinuous, or to have a discontinuity at that point.
Vertical, horizontal and oblique or slant asymptotes
A line whose distance from a curve decreases to zero as the distance from the origin increases without the limit is called the asymptote.
The definition actually requires that an asymptote be the tangent to the curve at infinity. Thus, the asymptote is a line that the curve approaches but does not cross.
Vertical asymptote
The line x = a is a vertical asymptote of a function  if  f (x) approaches infinity (or negative infinity) as x approaches a from the left or right.
Horizontal asymptote
The line y = c is a horizontal asymptote of a function if  f (x) approaches c as x approaches infinity (or negative infinity).
Oblique or slant asymptote
The line  y = mx + c  is a slant or oblique asymptote of a function  if  f (x) approaches the line as x approaches infinity (or negative infinity).
  - - - - - - -
The functions that most likely have asymptotes are rational functions.
So, vertical asymptotes occur when the denominator of the simplified rational function is equal to 0. Note that the simplified rational function has cancelled all factors common to both the numerator and denominator.
The existence of the horizontal asymptote is related to the degrees of both polynomials in the numerator and the denominator of the given rational function.
Horizontal asymptotes occur when either, the degree of the numerator is less then or equal to the degree of the denominator.
In the case when the degree (n) of the numerator is less then the degree (m) of the denominator, the x-axis
y = 0 is the asymptote.
If the degrees of both polynomials, in the numerator and the denominator, are equal then,  y = an / bm  is the horizontal asymptote, written as the ratio of their highest degree term coefficients respectively.
When the degree of the numerator of a rational function is greater than the degree of the denominator, the function has no horizontal asymptote.
A rational function will have a slant (oblique) asymptote if the degree (n) of the numerator is exactly one more than the degree (m) of the denominator that is if  n = m + 1.
Dividing the two polynomials that form a rational function, of which the degree of the numerator pn (x) is exactly one more than the degree of the denominator qm (x), then
pn (x) = Q (x) qm (x) + R      =>    pn (x) / qm (x) = Q (x) + R / qm (x)
 where, Q (x) = ax + b is the quotient and R / qm (x) is the remainder with constant R.
The quotient Q (x) = ax + b represents the equation of the slant asymptote.
As x approaches infinity (or negative infinity), the remainder R / qm (x) vanishes (tends to zero).
Thus, to find the equation of the slant asymptote, perform the long division and discard the remainder.
The graph of a rational function will never cross its vertical asymptote, but may cross its horizontal or slant asymptote. 
  - - - - - - - 
Example:  Given the rational function sketch its graph.
Solution:  The vertical asymptote can be found by finding the root of the denominator,
                      x + 1 = 0       =>       x -1  is the vertical asymptote.
The horizontal asymptote is the ratio of their highest degree term coefficients since the degree of polynomials in the numerator and denominator are equal,
  is the horizontal asymptote.
The graph of the given rational function is translated equilateral (or rectangular) hyperbola shown below.
The rational function of the form
can be rewritten into
so  
where, x0 and y0 are asymptotes and k is constant.
 
Therefore, values of the vertical and the horizontal asymptote correspond to the coordinates of the horizontal and the vertical translation of the source equilateral hyperbola  y = k/x, respectively.
Example:  Given the rational function   sketch its graph.
Solution:  The vertical asymptote can be found by finding the root of the denominator, 
                      x + 2 = 0       =>      x -2  is the vertical asymptote.
Since the degree of the numerator is exactly one more than the degree of the denominator the given rational function has the slant asymptote.
By dividing the numerator by the denominator
obtained is the slant asymptote  y = x 
and the remainder  3/(x + 2) that vanishes as x approaches positive or negative infinity.
 
Stationary points and/or critical points
The gradient of a curve at a point on its graph, expressed as the slope of the tangent line at that point, represents the rate of change of the value of the function and is called derivative of the function at the point, written dy/dx or f '(x)
At points of the graph where function changes from increase to decrease, the slope of the tangent line changes from positive to negative values respectively, passing through zero value. 
The points of the graph of a function at which the tangent lines are parallel to the x-axis, and therefore the derivative at these points is zero, are called the stationary points
There are three different types of stationary points: maximum points, minimum points and points of horizontal inflection. On the above graph the stationary points are denoted as, PMax, I and Pmin.
The graph reaches a local (or relative) maximum when gradient changes from positive through zero to negative.
The graph reaches a local (or relative) minimum when gradient changes from negative through zero to positive.
A local maximum is a value of the function greater than any adjacent value, i.e., in its immediate area it is the highest point, but it may not be the greatest value of the function over its whole range.
The endpoints of intervals of monotonicity are places where function stops increasing and starts decreasing or vice versa.
A function f (x) can change from increasing to decreasing (or vice versa) at values where f '(x) = 0 or f '(x) is undefined.
Note that these are only potential places where the graph can change from increasing to decreasing (or vice versa) since it is possible that the function may not change at those values, as for example at the point xI (where f '(x) = 0), in the above figure or, as in case of the rational functions from the above two examples, at the vertical asymptotes (where  f '(x) is undefined).
If f (x) is defined at x = c and either f '(c) = 0 or  f '(c) is not defined, then x = c is called a critical value of the function f (x), and its point (c, ƒ(c)) is called a critical point.
Therefore, a critical point may be a local maximum, a local minimum, or neither.
The critical point is neither a maximum nor a minimum if the function does not change from increasing to decreasing (or vice versa) at the critical point, as at the point xI in the above figure.
For a given function ƒ and point (c, ƒ(c)), the derivative of ƒ at x = c is the slope of the tangent line through the point (c, ƒ(c)), i.e.,  f '(c) = tan at .
The function value f (c) in the right figure is defined and the derivative at x = c is undefined, therefore the point (c, f (c)) is a critical point.
As f (x) is increasing before x = c and decreasing after x = c, the point (c, f (c)) is a local maximum.
Turning points (extremes, local or relative maximums or minimums)
A stationary point at which the gradient (or the derivative) of a function changes sign, so that its graph does not cross a tangent line parallel to x-axis, is called the tuning point.
Thus, a turning point is a critical point where the function turns from being increasing to being decreasing (or vice versa), i.e., where its derivative changes sign. 
A local (or relative) maximum is a point where the function turns from being increasing to being decreasing, i.e., where its derivative changes sign from positive to negative.
Notice that, as we travel through the maximum turning point from left to right, the derivative (the slope of the tangent to the curve) is decreasing, i.e.,  f '(x) changes from positive through zero to negative as x increases.
Thus, if the derivative of a function is decreasing over an interval, the graph of the function is concave down.
A local (or relative) minimum is a point where the function turns from being decreasing to being increasing, i.e., where its derivative changes sign from negative to positive.
Thus, as we travel through the minimum turning point from left to right, the derivative is increasing, i.e.,  f '(x) changes from negative through zero to positive as x increases.
Therefore, if the derivative of a function is increasing over an interval, the graph of the function is concave up.
Inflection points and intervals of concavity
A point on a curve at which it crosses its tangent, and concavity changes from up to down or vice versa, is called the point of inflection, as shows the above figure.
The graph is concave up on an open interval where the slope increases and concave down on an open interval where the slope decreases.
Therefore, the points on a curve that join arcs of opposite concavity are points of inflection.
If the gradient of the function does not change sign at the stationary point, then it is a point of horizontal inflection. 
Symmetry of a function, parity - odd and even functions
A function f that changes neither sign nor absolute value when the sign of the independent variable is changed is even, so that,   f (x) =  f (-x).
Therefore, the graph of such a function is symmetrical with respect to the y-axis, as is the graph shown in the left figure below.
The graph of an even function. The graph of an odd function.
  A function  f  that changes sign but not absolute value when the sign of the independent variable is changed is odd, so that,  f (x) = - f (-x). That is, for each x in the domain of  f,   f (-x) = -  f (x).  
Therefore, the graph of such a function is symmetrical with respect to the origin, as is the graph shown in the right figure above.
Transformations of original or source function
How some changes of a function notation affect the graph of the function
Some changes in a function expression (or an equation/formula) do not affect the shape or the form of the graph of the original or the given function yf (x).
Such changes include use of geometrical transformations to the graph of the function, like translations (or shifts) of the graph of the original function in the direction of the coordinate axes, or its reflection across the axes.
Translations of the graph of a function
The graph of a translated function    yf (x - x0)    is obtained 
translating (shifting) the graph of its original or source function  yf (x) horizontally by x0 units to the right.
The graph of a translated function   y f (x) + y0  or  y - y0 f (x)
is obtained translating (shifting) the graph of its original function yf (x) vertically by y0 units up.
The graph of a translated function
yf (x - x0) + y0    or    y - y0 f (x - x0)
is obtained translating (shifting) the graph of its original function yf (x) in both directions of the coordinate axes, horizontally by x0 units to the right and vertically by y0 units up.
Example:  Draw the graphs of the given three quadratic polynomials,
a)   y = x2 + 4x + 4  = (x + 2)2,       b)   y = x2 - 3     and    c)   y = x2 + 4x + 1  or   y + 3 = (x + 2)2
as the translations of the same source quadratic y = x2.
a)   y  = (x + 2)2 b)   y = x2 - 3 c)    y + 3 = (x + 2)2
Reflections of the graph of a function - changing polarity of variables
Change of the sign of the independent variable of a function, denoted as y = f (-x), reflects the graph of the given (original) function y f (x) across the y-axis.
Change of the sign of the function, denoted as  y = - f (x), reflects the graph of the given function y = f (x) across the x-axis.
Changes of the signs of both, independent variable and the function, denoted as y - f (-x), reflect the graph of the given function y = f (x), across the y-axis and the x-axis.
Example:  Given quadratic polynomial  y = f (x) = x2 + 4x + 1  or   y + 3 = (x + 2)2,  transform to:
                a)   y = f (-x),     b)   y = - f (x)     and     c)  y - f (-x), and draw corresponding graphs.
Solution:  a)   y = ƒ(-x) = (-x)2 + 4(-x) + 1 = x2 - 4x + 1              or     y + 3 = (x - 2)2
                b)   y -ƒ(x) =  -(x2 + 4x + 1) = -x2 - 4x - 1             or     y - 3 = -(x + 2)2
                c)   y -ƒ(-x) =  -((-x)2 + 4(-x) + 1) = -x2 + 4x - 1   or     y - 3 = -(x - 2)2
a)   y + 3 = (x - 2)2 b)   y - 3 = - (x + 2)2
c)   y -x2 + 4x - 1   or     y - 3 = - (x - 2)2
Pre-calculus contents D
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