Question

What is the reciprocal function?

Answer 1

Hint: The reciprocal function as the name suggests gives us the reciprocal of any fundamental function. The most common form of reciprocal function that we observe is, $y=\dfrac{k}{z}$, where ‘k’ is any arbitrary real number and ‘z’ is any variable. This implies that the reciprocal function has a constant in a numerator and a function or a variable in the denominator.

Complete step by step answer:
We will first generalize a reciprocal function and see some examples of reciprocal function.
The general form of a reciprocal function is given by the following expression:
\[\Rightarrow f\left( x \right)=\dfrac{a}{g\left( x \right)}+k\]
Here, in the above written function, we have:
$f\left( x \right)$ is the reciprocal function.
‘a’ is any arbitrary constant
$g\left( x \right)$ is any function. And,
‘k’ is also any real constant
For example:
$\begin{align}
  & \Rightarrow f\left( x \right)=\dfrac{2}{{{x}^{2}}} \\
 & \Rightarrow g\left( x \right)=\dfrac{4}{x+4} \\
 & \Rightarrow h\left( x \right)=\dfrac{1}{x-7}+4 \\
\end{align}$
This is the standard representation of a reciprocal function. Now, we will see how to convert a given function, [say $f\left( y \right)$] into a reciprocal function.
The reciprocal of a number is calculated by dividing the 1 with the number. Similarly, this can be used to calculate the reciprocal function of any fundamental function. This can be done by dividing 1 by the fundamental function. Mathematically, this can be written as:
Reciprocal function of $f\left( y \right)$ is equal to $\dfrac{1}{f\left( y \right)}$.
For example:
The reciprocal function of $\sin x$ is $\dfrac{1}{\sin x}$. This is equal to $\cos ecx$.
The reciprocal function of ${{e}^{x}}$ is equal to $\dfrac{1}{{{e}^{x}}}$. This is equal to ${{e}^{-x}}$.
Hence, we saw the definition of reciprocal functions with their examples.

Note: The reciprocal function is also known by the name of multiplicative inverse of a function. Also, for a given function to be called a reciprocal function, the numerator must always be a constant. For example: $f\left( x \right)=\dfrac{g\left( x \right)}{h\left( x \right)}$ is not a reciprocal function if $g\left( x \right)$ is not a constant.