Solve applications involving function composition

Ontario Grade 12 Mathematics

Characteristics of Functions

Follow a real-world process through two functions, in the correct order

Many real situations involve a quantity being changed in stages. For example, a mass of fruit may first be converted into a volume of juice, and that volume may then be used to calculate earnings. Function composition describes this process: the output of one function becomes the input of another. In this lesson, you will organize the stages, write a composite function, and use it to answer an application question.

What you will learn

Identify the input and output of each function in an application.
Represent a two-stage process using function composition.
Evaluate a composite function and explain what the result means.
Solve an application question by choosing and using the correct order of functions.

1. Prerequisite bridge: inputs, outputs, and function rules

A function is a rule that assigns an output to each allowed input. The input is the quantity you begin with. The output is the quantity the rule produces. For example, if a machine makes 3 labels for every box, then the number of labels depends on the number of boxes.

Function notation names the rule and shows its input. In

g(x)=0.72x

, the rule

g

takes an input

x

and multiplies it by

0.72

. If

x

is a mass in kilograms, then

g(x)

could represent a volume in litres. The units help explain what each value means.

To evaluate a function, replace its input variable with the given value and calculate. For instance, evaluating

g(10)

means finding the output of

g

when its input is

10

. When two rules are used in sequence, the first output must have the right units to serve as the second input.

g(10)=0.72(10)

A function connects an input to an output.
Use units to track what each input and output represents.
The output of one function can become the input of another.

2. Plain language and representations of composition

Function composition means applying one function and then applying another to the result. The notation

f(g(x))

means: start with

x

, find

g(x)

, and use that answer as the input to

f

. The function written inside the parentheses is applied first.

A useful way to organize an application is with a process chain. Write the starting quantity, then the first change, then the next change. A table can make the stages and their units clear. Only after the order is clear should you write the composition in symbols.

The order matters because the two functions may use different units or represent different steps. If one function changes kilograms into litres and another changes litres into earnings, the mass must be converted before the earnings rule can be used. Reversing the functions would feed the wrong kind of quantity into a rule.

If the rules are given by formulas, substitution can produce a single rule for the whole process. In

f(g(x))

, replace the input of

f

with the expression for

g(x)

. This combined rule is useful when you need to calculate the final result for a particular starting value or determine what starting value gives a desired result.

f(g(x))

In $f(g(x))$ , apply $g$ first and then $f$ .
A process chain or table can clarify the order before you use symbols.
Check that each function receives a quantity with suitable units.

3. Model an application and respect its domain

A mathematical model is a set of rules or equations used to represent a situation. In a composition application, name what each rule does. State its input and output, including units when they are given. This prevents a formula from becoming a collection of unexplained numbers.

The domain of a function is the set of input values allowed by its rule and context. In an application, the context may limit the input. For example, a mass cannot be negative. Also, a rule may only make sense over a stated range. A composite model is meaningful only when the output of the first function is an allowed input of the second.

When a question asks for a final amount, evaluate the functions in order. When it asks what starting amount is needed for a target result, set the composite output equal to the target and solve for the starting input. Check that the answer is allowed by the situation, and explain its meaning with units.

Do not assume a process is linear unless the problem gives a linear rule or enough information to establish one. In this lesson, use the supplied function rules as models. The goal is to connect the stages correctly and interpret the result, not to invent a rule that the situation does not support.

y=f(g(x))

Define what each function represents before composing it.
Check input restrictions and units at each stage.
For a target-output question, solve the composite model for its starting input.

4. A reliable method for application questions

Read the question and identify the quantity it gives and the quantity it asks for. Name each intermediate quantity as well. Then arrange the stages in the order the situation describes.

Write the functions with their meanings and units. Use the output of the first stage as the input of the next. If useful, draw arrows or make a small table. Next, write the composition and either evaluate it or solve an equation involving it.

Finally, interpret your numerical answer in a complete statement. Include units and, where appropriate, a sensible level of precision. A correct calculation does not fully answer an application question unless the result is connected back to the situation.

x\longrightarrow g(x)\longrightarrow f(g(x))

Identify the starting quantity, intermediate quantity, and requested result.
Compose the functions in the order the process occurs.
State the result in context and check that it is reasonable.

Stages in the juice-stand process

Stage	Input	Rule	Output
Convert fruit to juice	Mass in kilograms, $x$	$g(x)=0.72x$	Volume in litres, $g(x)$
Calculate net earnings	Volume in litres, $L$	$h(L)=3.50L-12$	Earnings in CAD, $h(L)$
Combine the stages	Mass in kilograms, $x$	$h(g(x))$	Earnings in CAD

Worked example

Finding the mass needed for a target earning

A small juice stand processes fruit. Each kilogram of fruit produces

0.72

litres of juice. Let

g(x)=0.72x

, where

x

is the mass of fruit in kilograms and

g(x)

is the juice volume in litres. The stand's net earnings, in CAD, from

L

litres are modelled by

h(L)=3.50L-12

. How many kilograms of fruit are needed for net earnings of CAD 50?

Identify the stages
The starting quantity is fruit mass in kilograms. The first rule converts that mass to juice volume in litres. The second rule uses litres to calculate net earnings. Therefore, apply $g$ before $h$ .
$x\longrightarrow g(x)\longrightarrow h(g(x))$
Write the composition
The earnings rule takes litres as its input, so put the volume expression $g(x)$ in place of $L$ . This makes the composite rule give earnings directly from the starting mass.
$h(g(x))=3.50(0.72x)-12$
Set the output to the target
The question asks for the mass that produces CAD 50 in net earnings. Set the composite earnings equal to $50$ and solve for $x$ .
$3.50(0.72x)-12=50$
Solve for the input
Add $12$ to both sides, then divide by $3.50(0.72)$ . The answer is about $24.6$ kilograms. The decimal is suitable because the model allows a mass to be measured to part of a kilogram.
$x=\frac{62}{2.52}\approx24.6$

Answer: The stand needs about 24.6 kg of fruit to reach net earnings of CAD 50, according to the model.

Check: At about 24.6 kg, the volume is about 17.7 L. Applying the earnings rule gives about CAD 50. This confirms both the function order and the calculated input.

Common mistakes and how to avoid them

Writing

g(h(x))

because the letters appear in that order in a list.

Correction: Follow the real process. Fruit mass is first converted to litres, and litres are then used to calculate earnings, so the composition is

h(g(x))

.

Putting the original mass directly into the earnings rule.

Correction: The earnings rule expects litres, not kilograms. First convert the mass using

g

, then use that volume as the input to

h

.

Reporting a numerical answer without units or context.

Correction: State what the number measures and connect it to the question. Here the answer is about 24.6 kg of fruit.

Using a composite formula without checking whether its inputs fit the situation.

Correction: Check the units and any stated input restrictions at each stage. The first output must be an allowed input for the next function.

Lesson summary

Function composition represents a process with more than one stage.
In $f(g(x))$ , apply $g$ first; its output becomes the input of $f$ .
Use units, a process chain, or a table to keep the order clear.
For a target-result application, set the composite output equal to the target and solve for the starting input.
Interpret the answer in context and check that it is reasonable.

Check your understanding

Question 1

A function

p

converts minutes to seconds. A function

q

uses seconds to find a distance. Which expression gives distance from an input of minutes,

t

?

$p(q(t))$
$q(p(t))$
$p(t)+q(t)$
$q(t)-p(t)$

Show answer and explanation

q(p(t))

The minutes must first be converted to seconds using

p

. The seconds are then used by

q

, so the correct order is

q(p(t))

.

Question 2

Let

a(x)=2x

and

b(u)=u+5

. What is

b(a(4))

?

$13$
$18$
$10$
$9$

Show answer and explanation

13

First,

a(4)=8

. Then

b(8)=8+5=13

.

Question 3

A composite model is

r(s(x))

, where

s

outputs a volume in litres and

r

takes litres as its input. What should you check before using this model for a particular

x

?

Whether $r$ can take the output $s(x)$ as an allowed input
Whether $s$ takes its input from $r$
Whether the two function names begin with different letters
Whether the output of $r$ has the same units as $x$

Show answer and explanation

Whether

r

can take the output

s(x)

as an allowed input

The output of

s

must be an allowed input for

r

. Checking the units and any input restrictions helps confirm that the composition makes sense.

Key terms

Function: A rule that assigns an output to each allowed input.
Function composition: A process in which the output of one function is used as the input of another.
Composite function: A function formed by applying one function and then another, such as $f(g(x))$ .
Domain: The set of input values allowed for a function or model.
Model: A mathematical rule or set of rules used to represent a situation.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D2.6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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