Lesson 1.2

Function Notation and Evaluation

Transitioning from "y equals" to "f of x" isn't just a name change—it's a shift to precise input-output instructions.

Introduction

Prerequisite Connection: In Algebra, you evaluated expressions like when by plugging in the number—function notation simply formalizes this request.

Today's Increment: We introduce the symbol , read as "f of x," which tells you explicitly which variable is the "input" and what rule to apply to it. You will learn the strict procedural habit of Using Parentheses for every substitution.

Why This Matters for Calculus: In Calculus, you won't just plug in numbers; you'll plug in entire expressions like . Without master-level discipline in substitution, finding a derivative becomes algebraically impossible.

Explanation of Key Concepts

The Anatomy of Function Notation

f(input)=output

fThe Name: Just a label (like "Frank"). Can be , etc.
xThe Input: The "placeholder." It sits inside the parentheses.
yThe Output: The result of the calculation. is literally .

The Golden Rule of Substitution

When evaluating a function, NEVER just write the number. ALWAYS write empty parentheses first.

Step 1: The Skeleton

Replace every with .

Step 2: The Fill

Drop the input into the holes.

Worked Examples

Level: Basic

Example 1: Numerical Evaluation

Let . Find .

Step 1: Create the skeleton

Step 2: Substitute -3

Step 3: Simplify

Answer: 28

Level: Intermediate

Example 2: Variable Substitution

Let . Find .

Here, the "input" is an entire algebraic expression.

Step 1: Create the skeleton

Step 2: Plug in (2w + 1)

Step 3: Distribute and Combine

Level: Advanced (Calculus Prep)

Example 3: Difference Quotient Setup

Let . Evaluate and simplify .

Find f(x)

Find f(3)

Step 2: Assemble the Fraction

Step 3: Factor and Cancel

(for

)

This process of "canceling the zero-maker" is exactly how you will solve derivatives.

Common Pitfalls

Thinking "f(x)" means "f times x":
It does NOT mean multiplication. You cannot divide by f. is a single symbol representing the output.
Forgetting Parentheses with Negatives:
If , then . But if you write , your calculator will give you , which is wrong.

Real-World Application

Physics: Projectile Motion

Physicists use function notation to specify exactly when an event happens.

h(t) = height at time t
h(0) = initial launch height
h(3) = height 3 seconds after launch

Writing is much faster and more precise than writing "The height of the object three seconds after the experiment began was 50 meters."