How to calculate inverse functions

Inverse function

You can think of mathematical functions as a kind of "automaton": you throw something in on one side and get something else out on the other. At Functions give one value one and get one for it Function value. The Inverse function f-1 the function f does exactly that opposite.

definition
An inverse function is a mathematical function that assigns its argument to a function value. A function g is therefore the inverse function of a function f, if y = f (x), then x = g (y). In other words, if we were to apply f and then g to an argument x, we would get this argument again: f (g (x)) = x.

A function f only has an inverse function if for every y in the range of values, there is only one value of x in the range of definition, for which the following applies: f (x) = y. The inverse of a function is mostly called f-1 written and spoken "f inverse".

The relationship between function and inverse function can be explained using the following picture:

Suppose we have a function f (x) = x3 and want to know for which value of x our function f (x) has the value 64. We know, of course, that we can find this value using the cube root. So is . In general it can even be said that if then . Generally speaking: the cube root is the inverse function of the cubic function f (x) = x3.

Determine inverse functions

The inverse function of a function can be determined in three steps:

  1. Rewrite the function as y = f (x)
  2. Solve the new function for x
  3. At f-1To write (x) as a function of x, x and y must be swapped

example

Find the inverse function of f (x) = x3 - 5

First we write the function as

y = x3 - 5

Then we solve the function for x

x3 = y + 5

The last thing we do is swap x and y:

The inverse function is so

Not all functions have an inverse function

It is not fundamentally the case that every function also has a corresponding inverse function. If a function has two or more different function values ​​for a value of x, it is usually not possible to simply determine the inverse function. This can be determined graphically with a horizontal line. If you draw the function, then a horizontal line may only intersect the graph at one point. If it intersects the graph at several points, there is probably no inverse function.

A function that assigns only one value from the set of values ​​to each value of x is called injective function.

example

The inverse function of the trigonometric function f (x) = sin (x) is f-1(x) = asin (x). f (10π) = 0 but asin (0) = 0.

f (x) = sin (x)f (x) = asin (x)

Attention!

It is tempting to assume that the inverse of f (x) = x² is the function is. Even if is true for all x ≥ 0, this is no longer true for all x <0. If x becomes less than zero, the square root is no longer for negative values ​​in Are defined. The inverse function for values ​​of x <0 is therefore . A distinction must therefore be made between cases. All functions with even exponents have this problem.