Inverse functions

Logarithmic functions

Exponential and logarithmic functions
Exponential and logarithmic functions are mutually inverse functions
Inverse functions
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(ƒ(x)) = x    and    ƒ(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 = ƒ(y) and solve for y.  Or form the composition ƒ(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.
Exponential functions
 - Exponential function y = ex   <=>   x = ln y,   e = 2.718281828...the base of the natural logarithm.
The exponential function is inverse of the natural logarithm function, so that  eln x = x.
 - Exponential function y = ax   <=>   x = loga  y,   where a > 0 and a is not 1.
 The exponential function with base a is inverse of the logarithmic function, so that
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, so that
f (f -1(x)) = x  that is,  f (f -1(x)) = f (ax) =  loga(ax) = x.
The domain of  f (x) = ax is the set of all real numbers.
The range of the  f (x) = ax is the set of all positive real numbers.
If a > 1 then is an increasing function and if  0 < a < 1 then is a decreasing function.
The graph of the exponential function passes through the point (0, 1). The x-axis is the horizontal asymptote to the graph, as shows the above picture.
Logarithmic functions
 - 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.
 - Logarithmic function y = loga x    <=>    x = a y,   where a > 0, a 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 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, so that

The domain of f (x) = loga x is the set of all positive real numbers.
The range of f (x) = loga x is the set of all real numbers.
If a > 1 then ƒ is an increasing function and if  0 < a < 1 then ƒ is a decreasing function.
The graph of the logarithmic function passes through the point (1, 0). The y-axis is the vertical asymptote to the graph, as shows the above picture.
Translated logarithmic and exponential functions
Example:  Given translated logarithmic function  y - 2 = log3 (x + 1), find its inverse and draw their graphs.
Solution:  Exchange the variables and solve for y, that is  x - 2 = log3 (y + 1)  which gives  y + 1 = 3x - 2.
Note that         y - y0 = loga (x - x0)  represents translated logarithmic function to base a
and          y - y0 = a (x - x0)      represents translated exponential function with base a
where,  x0 and  y0  are the coordinates of translations of the graph in the direction of the coordinate axes.
Pre-calculus contents F