Inverse functions

In mathematics, if f is a function from a set A to a set B, then an inverse function for f is a function from B to A, with the property that a round trip A to B to A (or from B to A to B) returns each element of the initial set to itself. Thus, if an input x into the function f produces an output y, then inputting y into the inverse function produces the output x, and vice versa.

A function f that has an inverse is called invertible; the inverse function is then uniquely determined by f and is denoted by F^-1 .

Let f and g be two functions. If

f(g(x)) = x and g(f(x)) = x,

then g is the inverse of f and f is the inverse of g.

Some functions would not have inverse functions. For example, f(x) = x² . There are two numbers that f takes to 4. That is f(2) = 4, f(-2) = 4 .So the inverse of f should have take 4 to 2 and -2. Therefore f has no inverse.

A function f has an inverse if and only if when its graph is reflected about the line y = x, the result is the graph of a function (passes the vertical line test). But this can be simplified. We can tell before we reflect the graph whether or not any vertical line will intersect more than once by looking at how horizontal lines intersect the original graph.

Horizontal Line Test

Let f be a function.

If any horizontal line intersects the graph of f more than once, then f does not have an inverse.

If no horizontal line intersects the graph of f more than once, then f does have an inverse.

The property of having an inverse is very important in mathematics, and it has a name.

A function f is one-to-one if and only if f has an inverse.

Steps for finding the inverse of a function f.

• Replace f(x) by y in the equation describing the function.
• Interchange x and y. In other words, replace every x by a y and vice versa.
• Solve for y.
Replace y by f-1(x).

The inverse of a function has all the same points as the original function, except that the x's and y's have been reversed. This is what they were trying to explain with their sets of points. For instance, supposing your function is made up of these points: { (1, 0), (-3, 5), (0, 4) }. Then the inverse is given by this set of point: { (0, 1), (5, -3), (4, 0) }.

An invertible function is a function that can be inverted. An invertible function must satisfy the condition that each element in the domain corresponds to one distinct element that no other element in the domain corresponds to. That is, all of the elements in the domain and range are paired-up in monogomous relationships - each element in the domain pairs to only one element in the range and each element in the range pairs to only one element in the domain. Thus, the inverse of a function is a function that looks at this relationship from the other viewpoint. So, for all elements a in the domain of f(x), the inverse of f(x) , f^-1 (x)) satisfies:

f(a)=b implies f^-1 (b)=a

And, if you do the slightest bit of manipulation, you find that:

f^-1 (f(a))=a