# Transformations of the Reciprocal Function

## Key Questions

An element(point, line, plane or any other geometric, complex or whatever figure) is said to undergo a transformation when ever one or more of its properties are changed.

#### Explanation:

A transformation just a rule; its more like a function. It takes an object and returns that object's image.

Transformations are done using: functions, matrices, complex numbers etc.

What we call object can be a point, a line etc. The basic fact about all objects is that object haves properties.

For example: The point $A \left(3 , 2\right)$ has only the property of position(in a Cartesian coordinate system).

Once you change point $A$'s position to let's say $B \left(6 , 4\right)$ by a particular procedure, we say you have transformed point $A$ to be point $B$.

And in which case the object is $A \left(x , y\right)$ and the image is $B \left(6 , 4\right)$

Our transformation could be the matrix: $\left(\left.\begin{matrix}2 \text{ "0 \\ 2" } 0\end{matrix}\right.\right)$

Proof : Because $\left(\left.\begin{matrix}2 \text{ "0 \\ 0" } 2\end{matrix}\right.\right) \times \left(\left.\begin{matrix}3 \\ 2\end{matrix}\right.\right) = \left(\left.\begin{matrix}6 \\ 4\end{matrix}\right.\right)$

The basic reciprocal function is $\frac{1}{x}$

#### Explanation:

The graph looks like:
graph{1/x [-10, 10, -5, 5]}

• The reciprocal function is:

$f \left(x\right) = \frac{1}{x}$

It's graph is as following:

This is an example of asymptote.

Since $x$ can take all values except $0$ for $f \left(x\right)$ to be defined,
Domain: $R - \left\{0\right\}$, i.e., all real numbers except 0.
Range: $R - \left\{0\right\}$, i.e., all real numbers except 0.