# Inverse of a Function

## Related Calculator: Inverse Function Calculator

We already know that function is a rule that allows for every value of `x` to find corresponding value of `y`. But what if we are given rule, i.e. function, and value of `y` and want to find corresponding value of `x`?

In other words we want to find function that will express `x` in terms of `y`. Such function is called **inverse function** and denoted by `f^(-1)`. If `y=f(x)` then inverse function is `x=f^(-1)(y)`.

For example, if `y=f(x)=x^3` then `f^(-1)(27)=3` because `f(3)=27`.

Not all functions have unique inverse. Let's take function `f(x)=x^2`. Since `f(-2)=4` and `f(2)=4` then `f^(-1)(4)` is undefined because there 2 values that corresponds to 4, namely, 2 and -2. Therefore, inverse of `y=x^2` is multi-valued function. In this cases we will say that function doesn't have inverse, because inverse is multivalued function which we don't consider as function.

So, function will have inverse only when for different values of `x` we have different values of `y`.

**Definition**: function `f` is called **one-to-one** (or **injective**) if it never takes same value twice, i.e. `f(x_1)!=f(x_2)` whenever `x_1!=x_2` .

One-to-one functions are precisely the functions that possess inverse functions.

From this definition follows that `y=x^2` is not one-to-one, because `y(2)=y(-2)=4` , so different values have same output.

From another side `y=x^3` is one-to-one because `x_1^3!=x_2^3` whenever `x_1!=x_2` (different values can't have same cube). Thus, `y=x^3` has inverse.

Graphically we can determine whether function is one-to-one using horizontal line test.

**Horizontal Line Test.** A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Now, let's give more formal definition of inverse function.

**Definition**. Let `f` be a one-to-one function with domain `X` and range `Y`. Then its inverse function `f^(-1)` has domain `Y` and range `X` and is defined as follows: `f^(-1)(y)=x\ <=>\ f(x)=y` .

Note that domain of `f` equals range of `f^(-1)` and range of `f` equals domain of `f^(-1)`.

When we want to concentrate on inverse function, we will usually write `y=f^(-1)(x)`, i.e. we will interchange `x` and `y`, because we traditionally use `x` as independent variable.

Inverse function means that if we apply `f` to `x`, we will obtain `y`.

If we now apply `f^(-1)` to `y`, we will arrive back at `x`. And vice versa: applying `f^(-1)` to `y` and then applying `f` to result will return us to `y`.

As result we can say that function and its inverse cancel influence of each other.

Thus, there are two properties of inverse functions:

- `f^(-1)(f(x))=x` for any `x` from `X`.
- `f(f^(-1)(x))=x` for any `x` from `Y`.

**Example 1**. Given `f(1)=4`, `f(2)=3`, `f(5)=10`. Find `f^(-1)(3)` , `f^(-1)(4)` , `f^(-1)(5)` .

`f^(-1)(3)=2` because `f(2)=3` .

`f^(-1)(4)=1` because `f(1)=4` .

`f^(-1)(5)` is undefined because we are not given value of `x` for which `f(x)=5` .

Now let’s see how to compute inverse functions. Since we have `y=f(x)` then we need to express `x` in terms of `y` to obtain `f^(-1)` .

**Steps for finding inverse**:

- Write `y=f(x)`.
- Express `x` in terms of `y` (if possible). You will get `x=f^(-1)(y)` .
- To obtain `y=f^(-1)(x)` interchange `x` and `y`.

Actually we can first perform step 3 and then step 2, i.e. interchange `x` and `y`, and then express `y` in terms of `x`.

**Example 2**. Find inverse of `y=x^3-3`

- `y=x^3-3`
- `x=root(3)(y+3)`
- `y=root(3)(x+3)` is inverse.

**Example 3**. Find inverse of `y=(x+3)/(2x-5)`.

- `y=(x+3)/(2x-5)`
- `2xy-5y=x+3` or `x=(5y+3)/(2y-1)`
- `y=(5x+3)/(2x-1)` is inverse.

To check whether we correctly found inverse check is `f(f^(-1)(x))=x` holds.

Here `f(x)=(x+3)/(2x-5)` and `f^(-1)(x)=(5x+3)/(2x-1)` so `f(f^(-1)(x))=((5x+3)/(2x-1)+3)/(2(5x+3)/(2x-1)-5)=x`

Thus, we found inverse correctly.

Interchanging roles of x and y gives the ability to plot graph of inverse function based on graph of function `y=f(x)`. Since point `(x_0,y_0)` lies on graph of `f` then `(y_0,x_0)` lies on graph of `f^(-1)`. So, **graph of inverse is obtained by reflecting graph of function about line `y=x`.**

**Example 4**. Sketch graph of `y=sqrt(x-1)` and its inverse.

Domain of `y=sqrt(x-1)` is `x-1>=0` or `x>=1` . Range is `y>=0` .

This means that domain of inverse is `x>=0` and range is `y>=1`.

To find equation of inverse, we write `x` in terms of `y`: `y^2=x-1` or `x=y^2+1`. Now interchange `y` and `x`: `y=x^2+1`.

So, inverse is `y=x^2+1`.

Now, we draw graph of the function `y=sqrt(x-1)` on interval `[1,oo)` and then reflect it about line `y=x`.