D2.2 Model applications with combined functions

Ontario Grade 12 Mathematics

Characteristics of Functions

Use functions together to represent quantities and relationships in real situations

A function describes how one quantity depends on another. In an application, one function may not be enough to answer the question. An event organizer, for example, can model ticket revenue and event costs separately, then combine them to find profit. The new function represents the whole situation. In this lesson, you will connect function rules to context, choose an operation, and interpret the result.

What you will learn

Identify the input and output of a function in an application.
Choose an operation that matches a real relationship between functions.
Interpret a combined function using its units, domain, and context.
Use a combined function to answer a practical question.

1. Prerequisite bridge: read a function in context

A function rule connects an input to an output. The input is the value supplied to the rule. The output is the value the rule produces. For an event, the number of tickets sold could be the input, and the money collected could be the output.

A function may be written as

R(n)

. This means the output of function

R

depends on input

n

. The parentheses identify the input; they do not mean multiplication.

Name the quantities and their units before using a rule. If

n

counts tickets, then

R(n)

might be measured in dollars. The allowed inputs must also make sense. A ticket count cannot be negative and is usually a whole number.

R(n)=revenue from n

\text{ tickets}

Identify the input and output before using a function rule.
Check units and whether the input is sensible in the situation.

2. Choose an operation that matches the situation

A combined function uses two or more function outputs to model a new quantity. Choose the operation from the relationship described in the situation.

Add functions when amounts contribute to a total, such as equipment cost plus staffing cost. Subtract when one amount is taken from another. Profit is revenue minus cost.

Multiply when the situation calls for a product, such as price per item multiplied by the number of items. Divide when one amount is shared across another, such as total cost per person. The divisor cannot be zero, and the result must make sense in context.

A composition models a two-stage process. One function’s output becomes another function’s input. For example, a function could give distance travelled after a certain time, and a second function could give fuel cost for a trip of that distance. The units must fit: the distance output must be an acceptable input to the cost function.

The domain is the set of allowed input values. A combined function’s domain must respect the rules of its parts and the situation. A ticket model, for example, may only allow non-negative whole-number inputs.

(f+g)(x)=f(x)+g(x),\quad(f-g)(x)=f(x)-g(x),\quad(f\circ g)(x)=f(g(x))

Match addition, subtraction, multiplication, division, or composition to the stated relationship.
For composition, the first function’s output must be a suitable input for the next.
Respect function rules and real-world limits when choosing inputs.

3. Connect words, rules, and meaning

Suppose

R(n)

represents ticket revenue and

C(n)

represents total event cost when

n

tickets are sold. The statement “profit is revenue minus cost” tells us to subtract. The resulting function gives profit for each allowed ticket count.

Revenue and cost are both measured in dollars, so their difference is also measured in dollars. A positive profit means revenue is greater than cost. A negative profit means cost is greater than revenue. A profit of zero means the amounts are equal.

A table can help show how the same input is used in both functions before the outputs are combined. The table in this lesson uses ticket counts to display revenue, cost, and profit side by side.

P(n)=R(n)-C(n)

Use the same input in functions that describe the same situation at that input.
Interpret the combined output with its units and with the meanings of the original functions.

4. Use and check a model

Evaluating a combined function gives an output for a chosen input. Comparing the output with zero can show whether the situation has a surplus or a shortfall.

A break-even input is one where revenue equals cost. For a profit function, this is where the output is zero. Check that any answer is allowed by the model and makes sense in the situation.

A function rule is a model. It describes a situation under particular conditions, but it does not guarantee that every real outcome will match exactly. Use the model only for conditions and inputs it was designed to represent.

P(n)=0\iff R(n)=C(n)

Positive profit means a surplus; negative profit means a shortfall.
Break-even occurs when the profit function equals zero.
Check that a calculated input is allowed and realistic.

Ticket count and predicted profit

Tickets sold, $n$	Revenue, $R(n)$	Cost, $C(n)$	Profit, $P(n)$
0	CAD 0	CAD 250	−CAD 250
20	CAD 360	CAD 370	−CAD 10
21	CAD 378	CAD 376	CAD 2

Worked example

Finding an event’s profit and break-even count

A school event charges CAD 18 per ticket. Venue and setup costs total CAD 250, and each attendee adds CAD 6 in costs. Let

n

be the number of tickets sold. Define revenue and cost, combine them to model profit, find the profit from 40 tickets, and determine the first whole-number ticket count at which the model predicts a non-negative profit.

Define revenue and cost
Revenue is the ticket price multiplied by the number of tickets. Cost includes a fixed CAD 250 plus CAD 6 for each ticket. Both outputs are measured in dollars, so their difference can represent profit.
$R(n)=18n,\quad C(n)=250+6n$
Combine the functions
Profit is revenue minus cost. Substituting the two rules gives one function for the event’s profit at a chosen ticket count.
$P(n)=R(n)-C(n)=18n-(250+6n)=12n-250$
Evaluate at 40 tickets
Use $n=40$ because the question asks about 40 tickets. The positive output means revenue exceeds cost by CAD 230 at this ticket count.
$P(40)=12(40)-250=230$
Find the first whole-number count with non-negative profit
Break-even occurs when profit is zero. Solving gives a threshold between 20 and 21 tickets. Since the input must be a whole number, check the two neighbouring counts to identify the first one with non-negative profit.
$12n-250=0\quad\Rightarrow\quad n=\frac{250}{12}\approx20.83$

Answer: The model predicts a profit of CAD 230 at 40 tickets. Its break-even threshold is approximately 20.83 tickets, so 21 is the first whole-number ticket count at which the model predicts a non-negative profit.

Check: At 21 tickets, revenue is CAD 378 and cost is CAD 376, giving a CAD 2 profit. At 20 tickets, revenue is CAD 360 and cost is CAD 370, giving a CAD 10 shortfall. This confirms that 21 is the first whole-number count with non-negative profit.

Common mistakes and how to avoid them

Adding revenue and cost to find profit.

Correction: Profit is the amount left after costs are taken from revenue, so subtract cost from revenue.

Combining function outputs without checking their inputs.

Correction: Confirm that the functions describe the same situation at the same input value before combining them.

Assuming every calculated input is realistic.

Correction: Check the domain and context. Ticket counts, for example, must be non-negative whole numbers.

Reading a negative profit as a negative ticket count.

Correction: The input is the ticket count. A negative output means costs exceed revenue.

Composing functions without checking the units.

Correction: The first function’s output must be a meaningful input for the second.

Lesson summary

A function models how an output depends on an input.
Choose an operation that matches the relationship in the application.
Profit can be modeled as revenue minus cost.
In a composition, one function’s output becomes another function’s input.
Interpret combined-function outputs using their units, domain, and context.

Check your understanding

Question 1

For an event,

R(n)

is revenue and

C(n)

is cost. Which function represents profit?

$R(n)+C(n)$
$R(n)-C(n)$
$C(n)-R(n)$
$R(C(n))$

Show answer and explanation

R(n)-C(n)

Profit is the money remaining after costs are subtracted from revenue.

Question 2

A function

d(t)

gives distance in kilometres after

t

hours. A function

F(x)

gives fuel cost for a trip of

x

kilometres. Which expression gives fuel cost after

t

hours?

$d(F(t))$
$F(t)+d(t)$
$F(d(t))$
$F(t)d(t)$

Show answer and explanation

F(d(t))

The distance after

t

hours is

d(t)

. This distance is the input to the fuel-cost function, giving

F(d(t))

.

Question 3

A model gives a profit of negative CAD 15 for 30 tickets sold. What does this output mean?

The event sold negative 15 tickets.
The event has CAD 15 more in costs than revenue.
The event has CAD 15 more in revenue than costs.
The event broke even.

Show answer and explanation

The event has CAD 15 more in costs than revenue.

The input is 30 tickets. The negative output means costs exceed revenue by CAD 15.

Key terms

Input: A value supplied to a function.
Output: The value a function produces for an input.
Combined function: A function formed by using two or more function outputs together.
Composition: A combined function in which one function’s output is used as another function’s input.
Domain: The set of input values allowed for a function or model.
Break-even: The point in a model where revenue and cost are equal, so profit is zero.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D2.2. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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D2.2 · Model applications with combined functions

Ontario Grade 12 Mathematics

Use functions together to represent quantities and relationships in real situations

What you will learn

1. Prerequisite bridge: read a function in context

2. Choose an operation that matches the situation

3. Connect words, rules, and meaning

4. Use and check a model

Ticket count and predicted profit

Finding an event’s profit and break-even count

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MHF4U

About this lesson and its review