The next objective is to explore the relationships between graphs of f and . For this purpose it will be desirable to use x as the independent variable for both functions, which means we are comparing the graphs of y = f(x) and y = (x).
If (a, b) is a point on the graph y = f(x), then b = f(The). This is equivalent to the statement that The = (B) which means that (b, a) is a point on the graph of y = (x).
In summary, inverting the coordinates of a point on the f produces a point on the . Similarly invert the coordinates of a point on the graph of produces a point on the f. However, the geometric effect of inverting the coordinates of a point is to reflect that point on the line. y = x (figure 1), and then the graphs of y = f(x) and y = (x) are each other relative to this straight line (Figure 2). In short, we have the following result.
|If f have a inversethen the graphs of y = f (x) and y = (x) are reflections of each other in relation to the straight y = x; that is, each is the mirror image of the other with respect to that straight line.|
Increasing or decreasing functions have inverse
If the graph of the function f always increasing or always decreasing over the domain of fthen this graph can be cut at most once by any horizontal line and hence the function f must have an inverse.
One way to tell if the graph of a function is increasing or decreasing in a range is by examining the inclinations of its tangent lines. The graph of f should be increasing at any interval where f '(x)> 0 (since tangent lines have a positive slope) and should be decreasing at any interval where f '(x) <0 (since tangent lines have negative inclination). These observations suggest the following theorem.
|If the domain of f is a range in which f ' (x)>0 or in which f '(x)<0then the function f there is a inverse.|
The graph of f(x) = is always growing in , once
for all x. However, there is no easy way to solve the equation. y = for x in terms of y; even knowing that f There is an inverse, we cannot produce a formula for it.
NOTE. What is important to understand here is that our inability to find a formula for the inverse does not negate its existence; indeed, it is necessary to develop ways of finding properties of functions, which have no explicit formula to work with.Next: Logarithmic and Exponential Functions