Function Notation

What is a Function?
A function is a rule that assigns exactly one output to each input. In other words, for every input x, there is exactly one corresponding output y.

What is function notation?
Functions are given letter names.
The names are of the form f(x) which is read “f of x”. The letter inside the parentheses, usually x, stands for the domain set.

The entire symbol, usually f(x), stands for the range set.

The ordered-pair numbers become (x, f(x)).

The following diagram shows what is function notation. Scroll down the page for more examples and solutions of function notations.

Function notation uses the symbol f(x) to represent the output of a function f when the input is x.
f is the name of the function.
x is the input (independent variable).
f(x) is the output (dependent variable).

Example:
Given f(x) = x² + 3x – 1, find
a) f(1)
b) f(–1)
c) f(a)
d) f(x – 1)

Solution:
a) f(1) = (1)² + 3(1) – 1 = 3
b) f(–1) = (–1)² + 3(–1) – 1 = –3
c) f(a) = a² + 3a – 1
d) f(x – 1) = (x – 1)² + 3(x – 1) – 1
= x² – 2x + 1 + 3x – 3 – 1 = x² + x –3

Example:
Give g(x) = x² + 2, find
a) g(a + b)
b) g(x²)

Solution:
a) g(a + b) = (a + b)² + 2
= a² + 2ab + b² + 2
b) g(x²) = (x²)² + 2 = x⁴ + 2

Function Notation
Throughout mathematics, we find function notation. Function notation is a way to write functions that is easy to read and understand.
Functions have dependent and independent variables, and when we use function notation the independent variable is commonly x, and the dependent variable is f(x).
In order to write a relation or equation using function notation, we first determine whether the relation is a function.

Function Notation
How to use the function Notation?
A basic description of function notation and a few examples involving function notation.
Example:
If g(t) = t² - 2t + 1

1. g(3t)
   
2. 5g(-1)
   
3. solve g(t) = 1
   If f(x) = 5x -2, h(x) = x²
   
4. 3f(x)
   
5. f(2x - 1)
   
6. -f(a)
   
7. Solve f(-x) = h(x) + 2

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