Algebraic and transcendental functions
Transcendental functions - The graphs of transcendental functions
-  Exponential and logarithmic functions are mutually inverse functions
The graph of the exponential function
The graph of the logarithmic function
-  Trigonometric (cyclometric) functions and inverse trigonometric functions (arc functions)
Algebraic and transcendental functions
Elementary functions are,   Algebraic functions and Transcendental functions
Algebraic functions
· The polynomial function   f (x) =  yanxn + an-1xn-1 + an-2xn-2 + . . . + a2x2 + a1x + a0
y a1x + a0                                                                    - Linear function
y = a2x2 + a1x + a0                                                      - Quadratic function
y = a3x3 + a2x2 + a1x + a0                                       - Cubic function
y = a4x4 + a3x3 + a2x2 + a1x + a0                        - Quartic function
y = a5x5 + a4x4 + a3x3 + a2x2 + a1x + a0         - Quintic function
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 · Rational functions - a ratio of two polynomials - Reciprocal function - Translation of the reciprocal function,     called linear rational function.
Transcendental functions - The graphs of transcendental functions
· Exponential and logarithmic functions are mutually inverse functions
- Exponential function  y = ex  <=>  x = ln y,   e = 2.718281828...the base of the natural logarithm,
exponential function is inverse of the natural logarithm function, so that  eln x = x.
- Logarithmic function  y = ln x = loge x    <=>    x = e y,    where x > 0
the natural logarithm function is inverse of the exponential function, so that  ln(ex) = x.
- Exponential function  y = ax   <=>   x = loga  y,   where a > 0 and  a is not 1

 exponential function with base a is inverse of the logarithmic function, so that
- Logarithmic function y = loga x    <=>    x = a y,   where a > 0a is not 1 and x > 0
the logarithmic function with base a is inverse of the exponential function, so that  loga(ax) = x.
The graph of the exponential function  y = ax = ebxa > 0  and  b = ln a
The exponential function is inverse of the logarithmic function since its domain and the range are respectively the range and domain of the logarithmic function and
ƒ(f -1(x)) = x  that is,  ƒ(f -1(x)) = ƒ(ax) =  loga(ax) = x.
The graph of the logarithmic function  y = logaxa > 0  and  for  a = ey = logex = ln x
The logarithmic function is inverse of the exponential function since its domain and the range are respectively the range and domain of the exponential function and

--   Trigonometric (cyclometric) functions and inverse trigonometric functions (arc functions)
Trigonometric functions are defined as the ratios of the sides of a right triangle containing the angle equal to the argument of the function in radians.
Or more generally for real arguments, trigonometric functions are defined in terms of the coordinates of the terminal point Q of the arc (or angle) of the unit circle with the initial point at P(1, 0).
 sin2x + cos2x = 1
College algebra contents B