Explicit and Recursive Function Forms

Lesson 1 of 2: Direct Formulas and Step-by-Step Builds

In this lesson:

• Write explicit formulas — compute any output directly
• Write recursive definitions — build from the previous value
• Verify both forms generate the same sequence

Lesson 1 Learning Objectives Overview

By the end of this lesson, you will be able to:

1. Write an explicit expression giving directly in terms of
2. Write a recursive process defining in terms of
3. Translate between explicit and recursive forms for linear and exponential functions
4. Verify that both representations generate the same sequence of values
5. Explain the trade-off: explicit gives direct access; recursive builds step by step
6. Choose the appropriate form based on the question being asked

Recall: Function Notation, Forms, and Evaluation

• Notation: , , — substitute the input to get the output
• Linear: — e.g., , so
• Exponential: — e.g., , so

How Do You Find the 100th Term?

You know the sequence starts: 3, 7, 11, 15, 19, ...

• Each term is 4 more than the previous
• To find the 100th term... do you need terms 1 through 99 first?
• Or is there a smarter way?

Think about which approach you'd actually use.

Explicit Expressions Give Direct Output Access

An explicit expression gives directly in terms of .

• Compute any output without knowing previous outputs
• Examples: , ,
• Want the 100th term? Substitute . Done.

Explicit = direct access to any output

Plumber Charges as a Linear Explicit Formula

Scenario: A plumber charges $75 plus $40 per hour.

• Cost for 6 hours:
• No need to compute through first

Investment Growth as an Exponential Explicit Formula

Scenario: An investment of $1000 earns 6% per year.

• Value after 20 years:
• Direct substitution — no intermediate values needed

The power of explicit: skip straight to any value you need.

Quick Check: Identify the Explicit Formula

A gym membership costs $25 to join plus $15 per month.

Which explicit formula models total cost after months?

• A)
• B)
• C)

Think before the next slide...

The Power of Explicit Form

With an explicit formula, you have direct access to any output:

• Find in one step — no intermediate values
• Useful when the question asks for a specific, distant term
• Works best for: linear, exponential, polynomial, and many other types

When Each Value Depends on the Previous

Some processes are naturally step-dependent:

• Bank balance depends on last month's balance plus interest
• Population depends on last year's count plus new births
• Each new value is built from the previous one

Recursive processes capture this dependence directly.

Recursive Processes: Building from Previous Values

A recursive process has two required parts:

1. Initial value: $f(0) = $ some starting number
2. Recursion rule: $f(n) = $ [expression involving ]

You must compute sequentially:

Both parts are required — the rule alone is incomplete.

Plumber Cost in Recursive Form

• Initial: (base fee, zero hours)
• Rule: (each hour adds $40)

Investment Growth in Recursive Form

• Initial:
• Rule:

To find : must compute through first.

Some Contexts Are Naturally Recursive by Design

"Each generation produces twice as many offspring, but 10 don't survive."

Means the Previous Output, Not Input

,

Quick Check: Build the Recursive Sequence

For , , compute through .

Guided Practice: Write the Recursive Form

Scenario: An account holds $500. Each year the balance is multiplied by 1.04.

Write the recursive definition:

• Initial: $V(0) = $ ___
• Rule: $V(t) = $ ___ ___

What is the initial value? What operation happens each step?

Comparing Explicit and Recursive: When to Use Each

Practice: Write Both Forms from Context

1. "A tree is 4 feet tall and grows 2.5 feet per year."
2. "A bank account starts at $800 and earns 3% interest annually."
3. Given: , — write the explicit form.

Practice Answers: Both Explicit and Recursive Forms

1. Tree: ; Recursive: ,

2. Bank: ; Recursive: ,

3. ✓ — verify: ✓, ✓

Lesson 1 Key Takeaways and Watch-Outs

✓ Explicit expressions give direct access — compute any output in one step

✓ Recursive processes build step by step — initial value AND rule required

✓ Both forms define the same function — explicit for distant terms, recursive for step-dependent processes

Watch out: To find recursively, you must compute through — no shortcut

Watch out: The rule alone is incomplete — generates infinitely many sequences without a starting value

Watch out: means the previous OUTPUT, not

Coming Up: Procedures, Choices, and Translation

Lesson 2 of 2: Procedural Descriptions and Translation Between Forms

• Procedures: when neither formula nor simple recursion fits
• Translation: convert explicit recursive for linear and exponential
• Choosing: which form is best for which question?