Function Notation Formula

Being a critical part of mathematics, functions and function notation formula exists at the center of mathematical analysis studies. Function notation is a symbolic representation of a function. Function notation helps in describing lengthy functions in a simpler way and their representations help us to understand them in a much easier way. Let's understand the function notation formula.

What is Function Notation Formula?

A function is an operator which operates on an input variable and produces an output. Functions arise whenever one quantity depends on another. Let's look into the usage of the function notation formula to understand the relationship between the input and output variables of a function.

In general, a function is represented using the letter 'f'. Also, other lower case letters such as 'g' or 'h' are also used to represent the function. 'f' along with the input variable enclosed inside parentheses() where the input variable is usually represented as 'x'  forms the function notation formula.

Let's take an example to understand the function notation formula.

Consider the relation y = x² where x is any real number. This equation tells us that y is dependent on x, because y is the square of x. In technical terms, y is a function of x, and this is specified using function notation formula as follows:

y = f(x)  or f : X → Y

where,

• f denotes the function name
• x is an element from the domain set X
• y or f(x) is an element from the range set Y
• The arrow indicates the mapping of input to the output

In other words, x is the input variable producing an output y or f(x).

As we know that, y = x²  thus, our function notation formula will be:

f(x) = x²

Let's look into some examples to understand the application of the function notation formula.

Solved Examples Using Function Notation Formula

Example 1:  y is a function of x, and the function definition is given as follows:
\[y = f\left( x \right) = \frac{1}{{1 + {x^2}}}\]
Find the output values of the function for \(x = 0\), \(x =  - 1\) and \(x = \sqrt 2 \) using function notation formula.

Solution:

The function notation formula given is:

\(y = f\left( x \right) = \frac{1}{{1 + {x^2}}}\)

Thus, by substituting the values of x we have,

\(\begin{align}&f\left( 0 \right) = \frac{1}{{1 +{{\left( 0 \right)}^2}}} = \frac{1}{1} = 1\\&f\left( { - 1} \right) = \frac{1}{{1+ {{\left( { - 1} \right)}^2}}} = \frac{1}{2}\\&f\left( {\sqrt 2 } \right) =\frac{1}{{1 + {{\left( {\sqrt 2 } \right)}^2}}} = \frac{1}{{1 + 2}} =\frac{1}{3}\end{align}\)

Answer: Thus, f(0) = 1, f(-1) = 1/2 and f(√2) = 1/3

Example 2: A cone has a variable height h and a variable base radius r, but the sum of h and r is fixed. The cone is made of a material of density \(\rho \). Express the mass m of cone as a function of its height h.

Solution: 

The volume V of a cone is given by:

\(V = \frac{1}{3}\pi {r^2}h\)

Let the (fixed) sum of h and r be k. Thus, \(r = k - h\), and so:

\(V = \frac{1}{3}\pi h{\left( {k - h} \right)^2}\)

Using function notation formula, the mass of the cone can now we expressed as a function of h; m will be \(\rho \) times the volume V:

\(m = f\left( h \right) = \rho V = \frac{1}{3}\pi\rho h{\left( {k - h} \right)^2}\)

Answer: Thus, \(m = f\left( h \right) = \rho V = \frac{1}{3}\pi\rho h{\left( {k - h} \right)^2}\) is the required function.