# 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**.

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) = x^{3} 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) = x^{3}.

### Determine inverse functions

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

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

### example

Find the inverse function of f (x) = x^{3} - 5

First we write the function as

y = x^{3} - 5

Then we solve the function for x

x^{3} = 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.

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