Functions

 Function

What is a Function?

A function is a rule that assigns each input (x) to exactly one output (y).

Domain: All possible input values (x-values).

Range: All possible output values (the results).

Function Notation

We usually write functions as f(x).

$f(x) = 3x + 1$

or Mapping

$f\colon x\mapsto3x+1$

Functions or Relation

    (1)One-to-One Function

        In a one-to-one function, every unique input (x) maps to a unique output (y). No two x-values                share the same y-value.

        Characteristic: It passes both the Vertical Line Test (it is a function) and the Horizontal Line Test (it         is one-to-one).

        Restriction: These are the only types of functions that can have an Inverse Function (f^{-1}).

            Example: f(x) = 2x + 1.

    (2)Many-to-One Function

        In a many-to-one function, two or more different inputs (x) can lead to the same output (y).    

        Characteristic: It is still a valid function because each x only goes to one y. However, it fails the            Horizontal Line Test.

        Restriction: You cannot find a single inverse function for the entire domain. To find an inverse, you         must "restrict the domain" (e.g., only looking at positive x values).

            Example: f(x) = x^2. Both x = 2 and x = -2 result in y = 4.

    (3)One-to-Many Relation (Not a Function)

        In a one-to-many mapping, a single input (x) results in multiple possible outputs (y).

Evaluating a Function

To evaluate a function, simply substitute the given value into the equation in place of x.

Example:

If f(x) = 2x² - 3, find f(5).

f(5) = 2(5)² - 3

f(5) = 2(25) - 3

f(5) = 50 - 3 = 4

Composite Functions

A composite function is when you combine two functions. The output of the first function becomes the input of the second.

fg(x) means you put function g into function f.

Rule: Always solve from inside to outside.

Example:

If f(x) = x + 2 and g(x) = 5x:

fg(x) = f(5x) = 5x + 2

gf(x) = g(x + 2) = 5(x + 2) = 5x + 10

Inverse Functions: f⁻¹(x) - [UNDO]

The inverse function "undoes" the original function. It takes the output back to the original input.

How to find the inverse (The 4-Step Method):

(1)Set the function to y. (e.g., y = f(x) = 4x - 7)

(2)Rearrange the equation to make x the subject.

(3)Swap x and y.

(4)Write the final answer as f⁻¹(x).

Example:

Find the inverse of f(x) = (x + 5) / 2

y = (x + 5) / 2

2y = x + 5   

x = 2y - 5

y = 2x - 5

f⁻¹(x) = 2x - 5

Important Graphs and Rules

(1)The Reflection Rule: The graph of f⁻¹(x) is always a reflection of f(x) in the line y = x.

(2)Domain Restrictions (Undefined Functions):

    There are two main "danger zones" where a function becomes undefined:

            1-Division by Zero: The denominator of a fraction cannot be zero.

            Example: For f(x) = 5 / (x - 2), the domain is x ≠ 2.

            2-Square Roots of Negative Numbers: The value inside a square root symbol (√) must be greater             than or equal to zero.

            Example: For f(x) = √(x - 3), the value inside (x - 3) must be ≥ 0.

            Therefore, the domain is x ≥ 3.

            If you pick a number smaller than 3 (like x = 2), you get √(-1), which is not a real number.

(3)Function vs. Relation: Use the Vertical Line Test. If a vertical line touches a graph at more than one         point, it is NOT a function.

Tip for the Exam:

Don't confuse f²(x) with [f(x)]².

f²(x) is a composite function: ff(x).

[f(x)]² is simply the function's result squared.