Inverse functions

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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:

Inverse functions

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:

Inverse functions

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

ExampleExample

Therefore:

Example

Rearrange to get,

xy + 2x = y

2x = y(1 − x)

Example

This means that the inverse function is,

Example

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 x2 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 → x2, 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 → x2, 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) ≡ 3x, x ∈ R, and sketch the graph.

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

Therefore:

for f(x), y = 3x

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

f−1(x) ≡ log3x

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:

Inverse functions

S-cool exclusive!!