Compose Functions

Building Functions: Combining Functions in a New Way

In this lesson:

• Chain two functions together:
• Find the domain of a composed function

Six Things You'll Be Able to Do

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

1. Define composition:
2. Evaluate compositions numerically, from tables
3. Evaluate compositions symbolically, by substitution
4. Find the domain of a composed function
5. Interpret compositions in dependency chains

Does Temperature Depend on Time?

You already know: height depends on time, and temperature depends on height.

So... does temperature depend on time?

Think about a weather balloon rising through the sky.

Chase the Chain by Hand

Find the temperature at using both tables.

You Just Composed Two Functions

Height at : . Temperature at that height: .

We write this chain as — read "T of h of t."

The output of became the input of .

One Chain, Two Steps: Inside Out

means: apply to first, then apply to that result.

• processes the input first
• processes 's output second
• Read from the inside out

The output of the inner function must work as an input for the outer function.

Watch the Chain Build, Step by Step

Time enters .

Height Comes Out of

gives height — the first link in the chain is complete.

Height Feeds Into , Temperature Comes Out

: time in, temperature out — the full composition.

Worked Example: The Full Balloon Chain

At : , then .

Step 1: Evaluate the inner function first —

Step 2: Feed that result into the outer function —

Quick Check: Chase This New Chain

Given and , find .

Chase the chain: what goes in, what comes out at each step?

Your Turn: Numeric Composition Practice

1. Find
2. Find

From Numbers to Symbols: Same Chase

, . Find , then .

Numeric: , so

Symbolic: , so

Same chase — now the "input" riding through the chain is a variable, not a number.

The Substitution Rule: Whole Expression In

To find : replace every in 's formula with the entire expression for .

• The whole inner expression moves in as one unit
• Use parentheses to keep it together

This is the chase you already know — written in algebra instead of arrows.

Worked Example: Substitute Step by Step

, . Find .

Step 1: Write with a blank:

Step 2: Fill the blank with

Step 3: Simplify

Same Pair, Other Order: Swap the Roles

, . Now find .

Step 1: Write with a blank:

Step 2: Fill the blank with

Step 3: Simplify

Predict, Then See the Order Matter

From the last slide: . Predict: will match?

Not the same. Composition is not commutative: .

Predict First, Then Read Inside Out

, . What is ? Pick before advancing.

• A.
• B.

Answer: B. is inside the parentheses — it runs first.

Second Example, Different Kind of Function

, . Find both compositions.

Different results — and, as we'll see soon, different domains too.

Your Turn: Predict, Then Compute Both Ways

, . Without computing: will ? Predict first.

1. Find
2. Find
3. Are they equal?

Substitute the entire inner expression as one unit — watch your parentheses.

A Chain That Computes, Then Breaks

, , at : , so — undefined.

Two Checks, Not One, for Domain

The domain of requires both:

1. must be in the domain of
2. must be in the domain of

Both checks must pass — the chain broke because check 2 failed, even though check 1 was fine.

Worked Example: Square Root Cascade

, . Find the domain of .

Check 1: accepts all reals — always passes

Check 2: need , so

Predict, Then Run Both Checks

, . Is in the domain of ? Predict first.

Check 1: in domain of — passes

Check 2: . Is in domain of ? No

Not in the domain, even though check 1 passed.

Worked Example: The Reciprocal Cascade

, . Find the domain of .

Check 1: accepts all reals — always passes

Check 2: need , so

Cascade vs. Intersection: Two Domain Rules

Arithmetic combination (, ): domain is the intersection of both domains — check both at once.

Composition : domain cascades — check , then check on 's output.

Solo Check: Commit Your Answer

, .

Is in the domain of ?

Write your answer down before we reveal it.

From Formulas to Real Chains

Composition also models real dependency chains:

• Building: "X depends on Y, which depends on Z" → write it as a composition
• Decomposing: a formula secretly hides two simpler steps — pull them apart

Building: Balloon Volume Over Time

Radius grows with time: . Volume of a sphere: .

Decomposing: Find the Hidden Chain

. What two simpler functions compose to give ?

Inner (runs first):

Outer (runs second):

Try a Second Valid Decomposition

. You've found one decomposition — inner , outer . Try a different one, then verify it recomposes to .

Decomposition 2: inner , outer — also recomposes to

Decomposition is not unique — choose the one most useful for your purpose.

Your Turn: A New Chain, No Scaffolding

Wind speed depends on altitude: . Fuel burn depends on wind speed: .

1. Write fuel burn as a function of altitude
2. Given , : evaluate at
3. is defined for wind speeds 0-60 — what about the domain?

Reveal: Fuel Burn Chain Resolved

Domain: valid only where falls between 0 and 60 — the wind-speed restriction cascades back onto which altitudes are allowed.

Five Mistakes Worth Watching For

means put into , not times

— order matters

Read inside out: first, second

Domain needs two checks: 's domain, then on

Decomposition is not unique — pick the useful split

What Composition Buys You Going Forward

Composition turns "this depends on that, which depends on something else" into one formula you can actually compute with.

Next: you'll run this chain in reverse to check whether two functions are inverses of each other — and in calculus, this same chaining is exactly what the chain rule differentiates.