Inverse Functions

Existence of an Inverse Function

Given a function f and an output y = f(x), we are often interested in finding what value or values x were mapped to y by f. For example, consider the function f(x) = x + 4. Since any output y = x +4, we can solve this equation for x to find that the input is x = y - 4. This equation defines x as a function of y. Denoting this function as f^−1, and writing x = f^-1(y). We see that for any x in the domain of f , f^−1( f(x) ) = f^-1(x+4) = x. Thus, this new function, f^−1, “undid” what the original function f did. A function with this property is called the inverse function of the original function.

 

Recall that a function has exactly one output for each input. Therefore, to define an inverse function, we need to map each input to exactly one output. let’s try to find the inverse function for f(x) = x^2. Solving the equation y = x^2 for x, we arrive at the equation x = ± y^0.5 . This equation does not describe x as a function of y because there are two solutions to this equation for every y > 0. The problem with trying to find an inverse function for f(x) = x^2 is that two inputs are sent to the same output for each output y > 0. We say a f is a one-to-one function if f(x1) ≠ f(x2) when x1 ≠ x2

 

※ Horizontal Line Test

A function f is one-to-one if and only if every horizontal line intersects the graph of f no more than once.

 

 

Finding a Function’s Inverse

We can now consider one-to-one functions and show how to find their inverses.Recall that a function maps elements in the domain of f to elements in the range of f. The inverse function maps each element from the range of f back to its  corresponding element from the domain of f. Therefore, to find the inverse function of a one-to-one function f, given any y in the range of f, we need to determine which x in the domain of f satisfies f(x)=y. Since f is one-to-one, there is exactly one such value x. We can find that value x by solving the equation f(x)=y for x. Doing so,we are able to write x as a function of y where the domain of this function is the range of f and the range of this new function is the domain of f.

 

※ Problem-Solving Strategy: Finding an Inverse Function

1. Solve the equation y=f(x) for x.

2. Interchange the variables x and y and write y=f^−1(x).

 

Graphing Inverse Functions

the graph of f^−1 is a reflection of the graph of f about the line y = x.

 

Restricting Domains

As we have seen, f(x) = x^2 does not have an inverse function because it is not one-to-one. However, we can choose a subset of the domain of f such that the function is one-to-one. This subset is called a restricted domain. By restricting the domain of f, we can define a new function g such that the domain of g is the restricted domain of f and g(x) = f(x) for all x in the domain of g. Then we can define an inverse function for g on that domain.

 

Inverse Trigonometric Functions

The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse.

why isn’t sin^−1(sin(π))=π? The issue is that the inverse sine function, sin^−1, is the inverse of the restricted sine function defined on the domain [−π 2 , π/2] . Therefore, for x in the interval [−π 2 , π/2],  it is true that sin^−1(sinx)=x.