Study sheet: Functions and Operations

📋 Course Outline

1. Relations and Functions
2. Testing Function Status
3. Domain and Range
4. Evaluating Functions
5. Operations on Functions
6. Composition of Functions

📖 1. Relations and Functions

🔑 Key Concepts & Definitions

• Relation : A set of ordered pairs (x, y).
• Function : A relation in which each value of x corresponds to exactly one value of y, so repetition of a domain element with different outputs is not allowed.

📝 Essential Points

• The ordered pairs {(0, 0), (-1, 1), (1, 1), (-2, 4), (2, 4)} form a function because no domain value is repeated with different outputs.

💡 Memory Hook

A relation pairs values; a function assigns each x exactly one y.

📖 2. Testing Function Status

★ Must-know

📌 An ordered-pair set is a function if every repeated domain value has the same corresponding output; it is not a function when one domain value is paired with different outputs.

📌 The vertical line test identifies a graph as a function when every vertical line intersects the graph at no more than one point; otherwise, the graph is not a function.

Further detail

• The Y-scan method identifies a non-function when:
  • y is missing
  • is inside the absolute value symbol
  • has an even exponent

💡 Memory Hook

Pairs → vertical line → Y-scan

📖 3. Domain and Range

🔑 Key Concepts & Definitions

• Domain : The set of all possible values of x.
• Range : The set of all possible values of y.

📝 Essential Points

• For the ordered pairs {(0, 2), (-1, -1), (2, 8), (1, 5), (-3, -7)}, the domain is {-3, -1, 0, 1, 2} and the range is {-7, -1, 2, 5, 8}.

💡 Memory Hook

Domain asks for x-values; range asks for y-values.

📖 4. Evaluating Functions

★ Must-know

• To evaluate a function at a given x-value, substitute that value for x in the function equation and simplify the resulting expression.

Further detail

• For f(x) = 3x - 2, evaluating at x = 0 gives f(0) = -2 and evaluating at x = 2 gives f(2) = 4.

💡 Memory Hook

Substitute → simplify → obtain f(x).

📖 5. Operations on Functions

📝 Essential Points

📐 Formula — For functions f and g with a common domain element x, addition is defined by $(f+g)(x)=f(x)+g(x)$.

📐 Formula — For functions f and g with a common domain element x, subtraction is defined by $(f-g)(x)=f(x)-g(x)$.

📐 Formula — For functions f and g with a common domain element x, multiplication is defined by $(f\cdot g)(x)=[f(x)][g(x)]$.

📐 Formula — For functions f and g with a common domain element x, division is defined by $(f/g)(x)=\frac{f(x)}{g(x)}$.

💡 Memory Hook

ASMD: Addition, Subtraction, Multiplication, Division

📖 6. Composition of Functions

★ Must-know

📐 Formula — The composition of functions f and g is defined by $(f\circ g)(x)=f(g(x))$.

Further detail

• If f(x)=x-1 and g(x)=x^2+1, then (f\circ g)(5)=f(g(5))=f(26)=25.

💡 Memory Hook

g acts first, then f: f(g(x)).

📊 Synthesis Tables

Function Identification Methods

Method	Function condition	Non-function condition
Ordered pairs	Each x has exactly one y	One x has different y-values
Vertical line test	Every vertical line intersects at most one point	A vertical line intersects more than one point
Y-scan method	None of the listed problematic y-patterns appears	y is missing, inside absolute value, or has an even exponent