Function Notation Formula, Definition, Solved Examples

Function Notation Formula: Function notation, a fundamental aspect of mathematics, lies at the core of studies in mathematical analysis. It serves as a symbolic representation of functions, aiding in the concise description of complex functions and facilitating a more accessible understanding of their operations.

What is the Function Notation Formula

A function represents an operation acting on an input variable to generate an output. Functions emerge whenever one quantity relies on another. Exploring the function notation formula allows us to grasp the connection between input and output variables in a function.

Typically denoted by the letter 'f', functions can also be represented by other lowercase letters like 'g' or 'h'. The function notation formula involves 'f' combined with the input variable contained within parentheses (). This input variable is commonly symbolized as 'x'.

For example, let's consider the relationship y = x ² , where x represents any real number. This equation illustrates that y relies on x, as y equals the square of x. In technical terms, y functions as a dependent variable of x, expressed through the function notation formula as follows:

y=f(x) or f:X→Y can be interpreted as:  f represents the function's name  x belongs to the set of elements in the domain X

y or f(x) belongs to the set of elements in the range Y

The arrow signifies the mapping from input to output

In simpler terms, x is the input variable generating an output  y or f(x).

Given y=x 2 , our function notation formula becomes: f(x)=x ²

Function Notation Formula Solved Examples

Example 1: Given the function notation formula: y=f(x)= 1/ 1+x ² ​

Substituting the values of f(0)= 1/ 1+(0) ² ​ = 1/ 1 ​ =1

f(−1)= 1/ 1+(−1) ² ​ = 1/ 1+1 ​ = 1 /2 ​

f( 2 ​ )= 1 / 1+( 2 ​ ) ² ​ = 1/ 1+4 ​ = 1/5 ​

Answer: Therefore, f(0)=1, f(−1)= 1/ 2 ​ , and f( 2 ​ )= 1/5 ​ .

Example 2: For a cone where the sum of height  h and base radius  r is fixed at  k, with r=k−h, the volume V of the cone is given by:

V=  1/3 ​ πh(k−h) ²

Using function notation, the mass  m of the cone is expressed as a function of  h, where  m is  ρ (density) times the volume  m=f(h)=ρV=  ​1/3 πρh(k−h) ²

Answer: Therefore, the required function expressing the mass of the cone as a function of its height is m=f(h)= 1/3 πρh(k−h) ² .

Example 3: Given a function f(x)=2x ² −3x+4, find the values for f(0), f(1), and f(2).

Solution: Substitute the values into the function:

f(0)=2(0) ² −3(0)+4=4

f(1)=2(1) ² −3(1)+4=3

f(2)=2(2) ² −3(2)+4=10

Example 4: Given the function g(t)= 1/t+3 ​ , determine the values for g(0), g(−1), and g(2).

Solution: Substitute the values into the function:

g(0)= 1/ 0+3 ​ = 1/ 3  ​

g(−1)= 1/ −1+3 ​ = 1/2 ​

g(2)=  1/ 2+3 ​ = 1/5  ​

Example 5: Consider a function h(x)= x +5. Calculate) h(4), h(9), and h(25).

Solution: Substitute the values into the function:

h(4)= 4 ​ +5 = 9

h(9)= 9 ​ +5 = 14

h(25)= 25 ​ +5 =30

Example 5: Consider a function h(x)= x +6. Calculate) h(4), h(9), and h(25).

Solution: Substitute the values into the function:

h(4)= 4 ​ +6 = 10

h(9)= 9 ​ +6 = 15

h(25)= 25 ​ +6 =31

Function notation is an important concept in mathematics that simplifies the representation and evaluation of functions, providing a concise and standardized way to describe the relationship between input and output variables in mathematical analysis. It enables clear communication and understanding of complex mathematical operations.

Explore Now Online Course of Class 9 Neev Fastrack 2024 and Class 10 Udaan Fastrack 2024 to enhance your Maths knowledge. and build a strong foundation.