A function is a type of operator that takes an input variable and provides a result. When one quantity is dependent on another, a function is created. An interesting property of functions is that each input corresponds to a single output. In other words, such an operator between two sets, say set A and set B is called a function if and only if it assigns each element of set B to exactly one element of set A.
Function Formula
The relation between input and output values of a particular function is given by the function notation formula. Generally, a function is depicted using the alphabet f in mathematics. The function notation formula is formed by the letters ‘f’ and the input variable contained by brackets, where the input variable is commonly represented as ‘x’. It is depicted as follows:
y = f(x) or f: A ⇢ B
where f is the name of the said function, x is an element from set A, f(x) is an element from set B and the arrow depicts the mapping from set A to set B.
Simply stated, x is the input value or variable which results in the output value, i.e. the range, depicted by y or f(x).
The list of function formulas can be extensively named as formulas for performing the various arithmetic operations across functions, and for performing the joined operations including at least two functions.
- The addition across two functions can be written as: (f + g)(x) = f(x) + g(x)
- The subtraction across two functions can be written as: (f – g)(x) = f(x) – g(x)
- The multiplication across two functions can be written as: (fg)(x) = f(x).g(x) or (αf)(x) = αf(x)
- The division across two functions can be written as: (f/g)(x) = f(x)/g(x)
Sample Problems
Problem 1: Represent y = x2 using function notation and find y at x = 12.
Solution:
Given: y = x2.
By using function notation, f(x) = x2.
Value of y at x = 12 means f(12).
So, f(12) = 122
⇒ f(12) = 144
Problem 2: Represent y = x3 + 4 using function notation and find y for x = 6.
Solution:
Given: y = x3+ 4.
By using function notation, f(x) = x3+ 4.
Value of y at x = 6 means f(6).
So, f(6) = 63 + 4
⇒ f(6) = 220
Problem 3: Represent y = x2 − 4x using function notation and find y for x = 3.
Solution:
Given: y = x2 − 4x.
By using function notation, f(x) = x2 − 4x.
Value of y at x = 3 means f(3).
So, f(3) = 32 − 4(3)
⇒ f(3) = −3
Problem 4: Represent y = 1 + x2 using function notation and find y for x =
Solution:
Given: y = 1 + x2.
By using function notation, f(x) = 1 + x2.
Value of y at x = √2 means f(√2).
So, f(√2) = 1 + (√2)2
⇒ f(√2) = 3
Problem 5: Find y for x = 7 in f(x) = 4x.
Solution:
Value of y at x = 7 means f(7).
So, f(7) = 4(7)
⇒ f(7) = 28
Problem 6: Find y for x = 9 for f(x) = x3.
Solution:
Value of y at x = 9 means f(9).
So, f(9) = 93
⇒ f(9) = 729