Combine Function Types With Arithmetic

HSF.BF.A.1.b

In this lesson:

• Add, subtract, multiply, and divide functions
• Build and take apart real-world models

What You Will Be Able to Do

By the end, you can:

1. Add, subtract, multiply, divide functions
2. Find the domain of a combined function
3. Build models from simple pieces (revenue, cost)
4. Explain what each piece of a model represents
5. Evaluate a combined function piece by piece
6. Connect cooling to the strategy

One Thermometer, Two Hidden Effects

A cup of coffee cools. The thermometer shows one number at a time.

But two things are happening:

• Room temperature — steady, unchanging
• The coffee's "extra heat" — fading on its own

Could two simple formulas, stuck together, predict that number?

Add the Outputs, Not the Inputs

Suppose and are two effects happening at the same moment.

What's their combined effect at that moment?

Compute it using only what you already know about arithmetic.

Naming What You Just Did

is just a name for "evaluate both, then add."

All Four Operations, One Pair

, :

Quick Check on Two New Numbers

If and :

What is ? What is ?

Work it out before advancing.

From Bare Numbers to Formulas

Same move as before — combine the expressions, then simplify.

Multiplying and Dividing the Same Pair

Same and :

Multiplying just raised the degree from 2 to 3.

Your Turn to Combine Two Functions

Find .

Find .

Try each one, then check your work on the next slide.

Checking Your Turn Against the Answer

Same Operation, Two Different Types?

Same operation, same — but the result's type depends on , not on the plus sign.

What If g Is Always Negative?

If is negative for every , is ever smaller than ?

Think about what subtracting a negative number does.

Two Real Effects, One Number

Room temperature: does it change minute to minute?

The coffee's extra heat: does that change?

Two separate, real effects — both happening at once, both worth their own formula.

Newton's Law of Cooling, Two Pieces

• : room temperature — constant
• : extra heat above the room — decaying exponential

Combining the Two Effects Into One

• The 68 is the baseline the temperature settles toward
• The is the shrinking gap above that baseline

Name the components, then combine with the matching operation.

Reading the Combined Model Over Time

At : the coffee starts at

As grows: the exponential term shrinks toward

So approaches — room temperature.

A Second Model: Revenue and Cost

• : revenue from selling units — 50 dollars per unit
• : cost to produce units — 1000 fixed plus 20 per unit

Two more real, separate effects. How should they combine?

Why Subtraction, Not Addition, for Profit

If you earn 500 dollars in revenue and spend 300 dollars in cost, is your profit 800 or 200?

Profit is what's LEFT after costs — always .

Finding the Break-Even Point From Profit

Set profit to zero and solve:

You can't sell a third of a unit — so break-even is at least 34 units.

Components Don't Need to Match

The cooling model adds a constant to a decaying exponential — totally different types.

Any functions can be combined — the result may be a brand-new type.

Same Formula, With the Labels Erased

You just built this. Without looking back, can you still name the two pieces just by reading the formula?

This is a recall check — the real test of decoding an unfamiliar formula comes next.

Reading a Model Backward Into Parts

Building a model: name components, then combine.

Decomposing a model: given only the combined formula, recover the components and their meaning.

Every term in a model was put there to represent something.

Decomposing a Manufacturing Cost Model

A manufacturing cost model. What might each part represent?

• : variable cost that grows with quantity (materials)
• : fixed cost that doesn't change (rent, equipment)

Decomposing a New Temperature Model

A different temperature model. What might each part represent?

• : ambient (room) temperature — constant
• : initial temperature excess, decaying over time

Decomposing a Three-Term Projectile Model

• : gravity pulling the object down
• : the push from initial upward velocity
• : the height it started from

Quick Check on the Profit Terms

In , what does each term represent?

Think back to the profit model — what does the 30x tell you? What does the 1000 tell you?

Find the Error in This Decomposition

A worked decomposition claims: "In , the term represents an upward push."

Is that right? What's the actual effect of a negative, squared term over time?

From Reading to a New Question

You've now built models forward and read them backward.

Next question: where is a combined model even valid? Not every input makes sense for every piece.

Can Profit's Input Be Negative?

Can be shirts? Can revenue exist at an where cost doesn't?

The combination can only be defined where both pieces are.

Domain Is the Overlap, Not the Coverage

To compute , you need BOTH AND to exist.

If either piece is undefined, the sum is undefined too.

Union Versus Intersection of Domains

is defined for . is defined for .

Division's Extra Domain Restriction Rule

Step 1 — intersect: and

Step 2 — for , also exclude : here, is never , so no extra points are removed

Domain Under a Real Constraint

If the factory's capacity caps production at 500 units, too.

Combined domain: — even though the algebra alone allows any .

Solo Commit: Predict This Domain

Predict the domain of .

Write your answer down before the next slide reveals it.

Full Transfer: A Brand-New Pair

Given and :

Compute , , , and .

State the domain of each.

No pre-filled steps this time — work through the full process on your own.

Which Cost Model Is Realistic?

Two proposed cost models for the same manufacturing scenario:

Model A: (fixed cost plus variable cost, added)

Model B: (fixed cost times variable cost)

Which one makes sense, and why?

Watch Out For These Five Mistakes

— add outputs, never nest

Adding doesn't always change type — multiplying tends to

Domain of is the overlap, not either's range

Profit is , never

Function types don't need to match to combine

What You Can Now Do

Complicated real-world quantities are rarely one formula.

They're usually two or three simple, well-understood functions — added or multiplied together — each one meaning something on its own.

Once you can see the pieces, you can build the model or take one apart.

Coming Up Next: Composing Two Functions

You can now spot the pieces inside a model and combine them side by side — each keeps its own meaning, lined up by a plus or minus sign.

But what if one piece's output feeds into the other as an input, instead?