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Logarithmic and Exponential Functions


Before we study these two types of functions, let's understand what inverse functions are.

Inverse functions

In common language, the term "inversion" conveys the idea of ​​a reversal. For example, in meteorology, temperature inversion is a reversal in the usual temperature properties of air layers; In music, an inversion is a recurring theme that uses the same notes in reverse order. In mathematics, the term inverse It is used to describe functions that are reverse to each other, in the sense that each undoes the effect of the other.

The idea of ​​solving an equation y = f (x) for x with a function of y, let's say x = g(y) is one of the most important ideas in mathematics. Sometimes solving this equation is a simple process; for example using basic algebra the equation

y = f (x)

can be resolved to x as a function of y:

x = g (y)

The first equation is better for calculating y if x is known, and the second is better for calculating x if y is known

The fundamental interest is to identify relationships that may exist between the functions f and g,when a function y = f(x) is expressed as x = g(y), or the other way around. For example, consider the functions and discussed above. When functions are composed in any order, one cancels the effect of the other meaning that

The first of these equations states that each output of a composition g(f(x)) is equal to the input, and the second states that each output of the composition f(g(y)) is equal to the input. Function pairs with these two properties are so important that there is specific terminology for them.

If the functions f and g satisfy both conditions

g(f(x)) = x for all x in the field of f

f(g(y)) = y for all y in the field of g

so we say that f and g are inverse functions. Also, we call f an inverse of g and g is an inverse of f.

Example

Confirm each of the following items.

(a) The inverse of

(b) The inverse of

Solution (a).

Solution (b).

NOTE. The result in the example should make sense to you intuitively, since the operations of multiply by 2 and multiply by In either order, they cancel each other's effect, just like cube raise and cube root operations.

Next: Reverse Functions Domain and Image