Understand Function Definition

Visualizing y = f(x)

Learning Objectives for This Deck

• Connect function notation to the coordinate plane.
• Graph functions by plotting .
• Identify domain and range from a graph.
• Recognize contextual restrictions on domain and range.

Hook: Seeing a Function Grow Over Time

Imagine a plant grows 2 inches every day.

• = days
• = height in inches

If we graph this, we can see the function's behavior across all time. The graph is the picture of the function's rule.

The Core Equivalence:

You spent years graphing equations like .

Now, we define .

The graph of the function is exactly the same as the graph of the equation .

Crucial Point: The -value and the value are the same quantity!

Reading Coordinates as Function Values

On a coordinate plane:

• The horizontal axis () represents the Input.
• The vertical axis () represents the Output — now called .

Every point on the graph is .

If you see the point , it means .

Building a Graph from a Table: Linear

Graphing : Parabola Example

Reading Function Values from a Graph

Finding Inputs: Working Backward from Outputs

Graphs also let us solve equations.

"If , what is ?"

1. Find on the -axis (the output side).
2. Move horizontally to reach the graph.
3. Read the -value directly below — that's the input.

The Vertical Line Test Explained

A graph represents a function only if every vertical line hits the graph at most once.

• Hits once: One input maps to one output — function. ✓
• Hits twice: One input maps to two outputs — NOT a function. ✗

Three Sources of Domain Restrictions

The domain isn't always all real numbers. It can be restricted by:

1. Context: Time can't be negative. Item counts must be whole numbers.
2. Algebraic Rules: Division by zero is undefined. Square roots of negatives are not real.
3. Explicit Specification: "Let the domain be ."

Restriction from Context: Water Tank

Scenario: A tank holds 100 liters and drains at 5 liters per minute.

• Domain: (time starts at 0; tank empties at ).
• Range: (level starts at 100, ends at 0).

Domain Restriction: Exclude Division-by-Zero Inputs

Consider .

If , the denominator equals zero — undefined.

Domain: All real numbers except .

The graph has a vertical asymptote (a "break") at .

Restriction from Algebra: Square Roots

Consider .

Square roots of negative numbers aren't real.

So , which means .

Domain: , written as .

Reading Domain from a Graph: Projection

The domain is the horizontal "shadow" the graph casts on the -axis.

Reading Range from a Graph: Projection

The range is the vertical "shadow" the graph casts on the -axis.

Domain and Range of a Line

(no context restrictions)

• The line extends left and right forever.
  • Domain: — all real numbers.
• The line extends up and down forever.
  • Range: — all real numbers.

Domain and Range of a Parabola

• The parabola spreads infinitely in both horizontal directions.
  • Domain: All real numbers.
• The lowest point (vertex) is at ; outputs only go up from there.
  • Range: , written as .

Domain and Range of a Semicircle

A semicircle (top half) with radius 3 centered at the origin.

• The semicircle only exists from to .
  • Domain: .
• It only reaches from to .
  • Range: .

Four Questions for Checking Domain and Range

When you see a new function, always ask:

1. Are there any denominators? (Set ≠ 0 for domain.)
2. Are there any square roots? (Set ≥ 0 for domain.)
3. Does the context restrict values? (Time, quantity, cost, etc.)
4. Does the graph show endpoints or a vertex?

Knowledge Check: Reading a Graph

If the point is on the graph of , what is the value of ?

• A) 5
• B) -2
• C) 3
• D) Unknown

Answer: B. The y-coordinate is the function value.

Knowledge Check: Identifying Domain Restrictions

What is the domain of ?

• A) All real numbers
• B)
• C) All real numbers except
• D)

Key Takeaways: Functions on the Coordinate Plane

• Graph = y = f(x): Every point encodes .
• Reading graphs: Find , travel to the curve, read .
• Domain: Horizontal shadow onto the -axis.
• Range: Vertical shadow onto the -axis.
• Three restrictions: context, algebraic rules, explicit specification.