D2.3 Determine properties of combined general functions

Ontario Grade 12 Mathematics

Characteristics of Functions

How operations and composition change a function’s domain, values, and graph

A new function can be made by combining two functions. For example, one function might describe a cost and another an amount produced. Their sum can describe total cost. To understand the new function, examine how its rule is built and which input values are allowed. This lesson focuses on properties of combined functions using Grade 12 algebra, tables, and graphs.

What you will learn

Build new functions by adding, subtracting, multiplying, dividing, or composing functions.
Determine the domain of a combined function from the domains of its component functions.
Use equations, tables, and graphs to investigate values, zeros, intercepts, and changes in a combined function.

1. Prerequisite bridge: inputs, outputs, and domains

A function assigns one output to each allowed input. The domain is the set of allowed input values. For example, if

f(x)=x^2

, then

f(3)=9

. The input is

3

, and the output is

9

.

Before combining functions, review two common domain restrictions. A denominator cannot equal zero. The expression inside a square root must be zero or positive when the function is real-valued. These restrictions still apply after functions are combined.

A graph shows input-output pairs visually. A table lists selected pairs. Both can help compare the component functions with the new function. A zero is an input where the output is zero. The

y

-intercept is the output when the input is zero, if zero is in the domain.

f(a)=b

Check the allowed inputs before using a function rule.
For a combined function, an input must be allowed wherever the component functions are used.

2. Plain language and symbols for combinations

Adding functions means adding their outputs for the same input. Subtracting functions means subtracting those outputs. Multiplying and dividing functions also use outputs from the same input. These operations create new functions.

Composition is different. To find

f(g(x))

, first use

x

as an input to

g

. Then use the output of

g

as the input to

f

. The order matters:

f(g(x))

and

g(f(x))

usually give different results.

The domain of a sum, difference, or product includes inputs allowed in both component functions. For a quotient, those inputs must also make the denominator function’s output nonzero. For a composition,

x

must be allowed in the inner function, and the inner output must be allowed as an input to the outer function.

When investigating properties, start with a specific question. To find a zero, set the combined output equal to zero and solve within the domain. To find a

y

-intercept, evaluate the combined function at zero if zero is allowed. A graph or table can help locate likely values, but the function rule can confirm them.

Rates of change describe how outputs change as inputs change. A numerical rate of change over an interval is the output change divided by the input change. A graph can show whether a combined function rises or falls over an interval; a table can provide numerical evidence. This is not the same as assuming the component functions’ rates simply add in every combination.

\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(f-g)(x)&=f(x)-g(x)\\(fg)(x)&=f(x)g(x)\\\left(\frac{f}{g}\right)(x)&=\frac{f(x)}{g(x)},\quad g(x)\ne0\\(f\circ g)(x)&=f(g(x))\end{aligned}

For composition, evaluate the inner function first.
A quotient’s denominator must be nonzero.
Use the combined rule, rather than guessing from one component, to test a property.

3. Multiple representations: rules, values, and graphs

Suppose a table gives values for two functions at the same inputs. For a sum, add the outputs in each row. For a product, multiply them. For a quotient, divide only when the second output is not zero. This makes a numerical representation of the combined function.

An equation shows the exact rule. It can reveal restrictions that a short table might miss. For example, a table may not include the input that makes a denominator zero, so the table alone cannot establish the full domain.

On a graph, the sum’s output at an input is the vertical total of the two function values, using their signed values. If one value is negative, it reduces the total. For a difference, subtract the second signed value. Products and quotients are also found from the values at matching inputs, not by simply adding or subtracting graph heights.

For composition, follow the input through two stages. A mapping diagram or table can show

x

going to

g(x)

and then to

f(g(x))

. This is useful when the algebra becomes complicated, but the domain still needs attention at both stages.

A combined function’s zeros and intercepts may differ from those of its components. For example, a product is zero when at least one factor is zero, provided the input is allowed. A sum is zero when the component outputs cancel. A quotient is zero when its numerator is zero and its denominator is not. (fg)(x)=0\iff f(x)=0 or g(x)=0

Tables give selected values; equations help check the full domain.
The signs of function values matter when adding or subtracting.
For a product, either factor can make the product zero; for a quotient, the denominator cannot be zero.

4. Apply the ideas and check the result

When you meet a combined-function question, name the operation first. Write its rule using the component functions. Then find the domain before solving for zeros, intercepts, or other properties. This order prevents an algebraic answer from being accepted when it is not an allowed input.

For composition, substitute the entire inner expression into the outer rule, using brackets to keep the order clear. Simplify only after the substitution. To check a result, choose an allowed input and evaluate it both from the combined rule and by applying the component functions in sequence.

A graph can support conclusions about where a function is positive, negative, rising, or falling. A table can support a numerical rate of change between two listed inputs. State the interval or inputs used, because a rate of change can differ over different intervals.

average rate of change =

\frac{F(b)-F(a)}{b-a}

, a\ne b

Identify the operation, write the rule, and check the domain.
Use substitution or a graph/table check to catch errors.
Describe graphical or numerical change over a stated interval.

How to build common combinations

Combination	What to do at input $x$	Domain check
Sum	Add the two outputs	Input must be allowed in both functions
Difference	Subtract the second output from the first	Input must be allowed in both functions
Product	Multiply the two outputs	Input must be allowed in both functions
Quotient	Divide the first output by the second	Both inputs must be allowed; the second output cannot be zero
Composition	Use the inner output as the outer input	Inner input and resulting outer input must both be allowed

Worked example

Build and analyze a composed function

Let

f(x)=x^2-1

and

g(x)=x-2

. Determine the rule and domain of

h(x)=f(g(x))

. Find its zeros and

y

-intercept, then check one value by applying the functions in sequence.

Identify the order
The notation $f(g(x))$ means that $g$ is applied first. Its output is then used as the input to $f$ .
$x\longmapsto g(x)\longmapsto f(g(x))$
Substitute the inner rule
Replace every input $x$ in the rule for $f$ with the whole expression $g(x)$ . Brackets show that the full inner output is being squared.
$h(x)=(x-2)^2-1$
Determine the domain
Both component rules are polynomials, so each accepts every real input. The output of $g$ is also allowed as an input to $f$ . Therefore the composition has no excluded real inputs.
$\operatorname{Dom}(h)=\mathbb{R}$
Find the zeros
Set the combined output equal to zero. The squared expression must equal one, so the expression inside the square is either one or negative one. Both resulting inputs are in the domain.
$(x-2)^2-1=0\quad\Rightarrow\quad x=1,\ 3$
Find the intercept
The $y$ -intercept occurs at input zero. Substituting zero into the combined rule gives an output of three.
$h(0)=(-2)^2-1=3$
Check by applying each function
Use input $x=4$ . First, $g(4)=2$ . Then evaluate $f(2)$ , which is three. This matches the value from the combined rule.
$g(4)=2,\qquad f(g(4))=f(2)=3=h(4)$

Answer:

h(x)=(x-2)^2-1

has domain all real numbers, zeros at

1

and

3

, and

y

-intercept

(0,3)

.

Check: The check confirms that the composition rule preserves the required order: apply

g

first, then

f

.

Common mistakes and how to avoid them

Finding

f(g(x))

by adding

f(x)

and

g(x)

.

Correction: Composition is a two-stage substitution. Find

g(x)

first, then use that expression as the input to

f

.

Keeping an input that makes the denominator of a quotient zero.

Correction: Exclude every input where the denominator function’s output is zero, even if simplifying the expression seems to remove the restriction.

Assuming zeros of a sum are the zeros of either component.

Correction: A sum is zero when the two outputs add to zero. They may cancel even when neither output is zero.

Treating a few table values as proof of the full domain or graph behaviour.

Correction: Use the function rules to identify restrictions. Use tables and graphs as evidence over the values or intervals shown.

Lesson summary

Combined functions are formed by adding, subtracting, multiplying, dividing, or composing function outputs.
Find the domain by checking the restrictions of every component and the operation used.
For composition, evaluate the inner function before the outer function.
Use equations to calculate exact properties and tables or graphs to support numerical or visual conclusions.

Check your understanding

Question 1

If

f(x)=x+1

and

g(x)=2x

, what is

(f\circ g)(3)

?

$7$
$8$
$9$
correctIndex

Show answer and explanation

7

First,

g(3)=6

. Then

f(6)=7

, so the composition has value

7

.

Question 2

For

q(x)=\frac{x+2}{x-4}

, which input must be excluded from the domain?

$-2$
$0$
$4$
correctIndex

Show answer and explanation

4

The denominator is zero when

x=4

. A quotient is undefined at that input.

Question 3

Suppose

f(2)=5

and

g(2)=-5

. What is

(f+g)(2)

?

$-10$
$0$
$10$
correctIndex

Show answer and explanation

0

Add the two outputs at the same input:

5+(-5)=0

.

Key terms

Combined function: A function formed by applying an operation to the outputs of one or more functions.
Composition: A function combination in which the output of one function becomes the input of another.
Domain: The set of input values for which a function is defined.
Zero: An input value that makes a function’s output equal to zero.
Average rate of change: The change in output divided by the change in input over an interval.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D2.3. 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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D2.3 · Determine properties of combined general functions

Ontario Grade 12 Mathematics

How operations and composition change a function’s domain, values, and graph

What you will learn

1. Prerequisite bridge: inputs, outputs, and domains

2. Plain language and symbols for combinations

3. Multiple representations: rules, values, and graphs

4. Apply the ideas and check the result

How to build common combinations

Build and analyze a composed function

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