Understand Function Definition

Visualizing y = f(x)

Learning Objectives

• 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: Visualizing Growth

Imagine a plant grows 2 inches every day.

• = days
• = height

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 Bridge:

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!

Coordinates:

On a coordinate plane:

• The horizontal axis () represents the Input.
• The vertical axis () represents the Output, which we now call .

Every point on the graph is .
If you see the point , it means .

Step-by-Step Graphing: Linear

Graph .

Plot these points and connect them. Every point on that line follows the rule of .

Step-by-Step Graphing: Non-Linear

Graph .

Notice how the outputs (-values) start high, go down to -4, and then go back up? That's the shape of the function!

Reading the Graph

To find , find on the -axis, move up to the graph, and read the -value. The graph is a visual lookup table.

Working Backward

Graphs also help us solve equations.

"If , what is ?"

1. Find on the -axis (the output).
2. Look left/right to find the graph.
3. Read the -value (the input) directly below it.

The Vertical Line Test

A graph represents a function only if every vertical line hits the graph no more than once.

• Hits once: One input = One output. (Function!)
• Hits twice: One input = Two outputs. (Error! Not a function!)

This is just the formal definition of a function shown visually.

Sources of Domain Restrictions

The domain isn't always "all real numbers." It is restricted by:

1. Context: Time can't be negative. People can't be fractions.
2. Algebraic Rules: You can't divide by zero. You can't take the square root of a negative (in the real numbers).
3. Explicit Choice: "Let the domain be {1, 2, 3}."

Case 1: Real-World Context

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

• Domain: Time starts at and stops at (when the tank is empty). .
• Range: Height starts at and stops at . .

Case 2: Math Rules (Division)

Consider .

If , the denominator is zero. The universe explodes (mathematically).

Domain: All real numbers except .

On a graph, you would see a "break" or an "asymptote" at .

Case 3: Math Rules (Square Roots)

Consider .

We can't take the square root of a negative number and get a real result.
So, must be .

Domain: .

Visualizing Domain: Projection

The Domain is the horizontal "shadow" the graph casts on the -axis. If the graph exists above or below an -value, that is in the domain.

Visualizing Range: Projection

The Range is the vertical "shadow" the graph casts on the -axis. It shows every height the function actually reaches.

Example: Domain/Range of a Line

(with no context)

• The line goes left and right forever.
  • Domain: or "All Real Numbers."
• The line goes up and down forever.
  • Range: or "All Real Numbers."

Example: Domain/Range of a Parabola

• The graph spreads wide forever.
  • Domain: All Real Numbers.
• The lowest point (vertex) is at . The graph goes up from there.
  • Range: .

Example: Domain/Range of a Semicircle

A semicircle with radius 3 centered at the origin.

• It only exists from to .
  • Domain: .
• It only exists from to .
  • Range: .

The Domain/Range Checklist

When analyzing a function, always ask:

1. Are there any denominators? (Check for zero)
2. Are there any square roots? (Check for negatives)
3. Does the context restrict the values? (Check for logic)
4. Does the graph show an endpoint or a peak?

Knowledge Check 1

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 2

What is the domain of ?

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

Summary

• Graph = y = f(x): The visual representation of the rule.
• Domain: The "width" of the graph (possible inputs).
• Range: The "height" of the graph (possible outputs).
• Context Matters: Math doesn't live in a vacuum—the real world sets the boundaries!