# Inverse functions

## Inverse functions

This area takes functions, but where we usually take values of x and look at the corresponding values of f(x), here we take values of f(x) and look at what value of x produces this.

**Inverse functions with one to one mapping**

**Let's look at an example:**

If we have the function, f: x → 3x, and the '**domain**' x ∈ {1, 2, 3}

**Then we see the mappings are for this function over these x values is:**

So now we have the mapping of x to f(x). But we can also go backwards and map the range values (i.e. the values produced by f(x)) to those in the domain (i.e. the values of x used).

So forward mapping gives |

1 → 3 |

2 → 6 |

3 → 9 |

And inverse mapping gives |

3 → 1 |

6 → 2 |

9 → 3 |

**Looking at the inverse mapping, the values produced can also be written as another function:**

x → x/3, where x → {3, 6, 9}.

This reverse mapping is a one-to-one mapping and is called the **inverse function** of f where f: x → 3x.

The symbol for any inverse is f^{−1}.

So, f^{−1} |
x → x/3, | x ∈ {3, 6, 9} |

is the inverse of f | x → 3x, | x ∈ {1, 2, 3} |

The relationship between the graphs of f and the inverse f^{−1} is shown in the diagram:

From the diagrams you can see that the transformation to get from f(x) to f^{−1}(x) is a **reflection in the line y = x.**

**This helps us to find the inverse of more complicated functions, and we do so by:**

Writing the equation as y = f(x).

Swapping the letters x and y. (This is the same as reflecting in the line y = x.)

Rearranging the formula into a new y = f(x). This is the inverse function.

**Example:**

Find the inverse function of

**Therefore:**

Rearrange to get,

xy + 2x = y

2x = y(1 − x)

This means that the inverse function is,

The above example had a '**one to one**' mapping (see lesson 1 - mapping). If you have a one to many mapping this causes complications.

This is because a single value of f(x) can be generated from many different values of x and this cannot be defined using a single inverse function. The way we can get around this is to set the **domain** (the range of x values the function can use), such that only one value of x will produce one value of f(x).

**This is quite a complex idea, so let's look at an example.
**

Using the function x^{2} the rearrangement gives us,

f^{−1}(x) = ±√x

This would then define a **one-to-many mapping** and therefore not give a function (as a function cannot be a one-to-many mapping).

Therefore,
f: x → x^{2},
x ∈ R
does not have an inverse function.

You can obtain the reverse mapping by only allowing x to take positive real numbers (or only negative real numbers).

So we have,
f: x → x^{2},
x ∈ R^{+}
which is a one-to-one mapping.

The reverse mapping only allows positive square roots in the range.

So, the inverse is, f^{−1} :x → √x, where
x ∈ R^{+}

As mentioned earlier, all you need to do to sketch the graph of the inverse function f^{−1} is **to reflect f in the line y = x.**

However, if *f* does not have an inverse, you will still be able to reflect the graph, but it will not represent the inverse function.

**Example:**

Find the inverse function of f(x) ≡ 3^{x}, x ∈ R, and sketch the graph.

To find f^{−1} we have to map values of 3^{x} back onto the values of x.

**Therefore: **

for f(x), y = 3^{x}

For f^{−1}(x), x = 3^{y}, and by taking logarithms we get,

f^{−1}(x) ≡ log_{3}x

The curves of the two function are the same, but the inverse function has been reflected in the line y = x. See the diagram below: