D2.1 Analyze sums, differences, products, and quotients of functions

Ontario Grade 12 Mathematics

Characteristics of Functions

Sums, differences, products, and quotients

A function assigns an output to each allowed input. You can think of a function as a rule or a machine: give it an input, and it returns an output. If two functions describe quantities that depend on the same input, you can combine their outputs. For example, if one function gives the cost of a ride and another gives the cost of a fee, their sum can give the total cost. This lesson focuses on analyzing those combined functions and their domains.

What you will learn

Determine how to combine two functions using addition, subtraction, multiplication, or division.
Write and simplify a rule for a combined function.
Identify which input values are allowed, including values excluded from a quotient.
Interpret a combined function in terms of the quantities represented by the original functions.

1. Prerequisite bridge: inputs, outputs, and domains

The input of a function is often written as

x

. The output of a function

f

is written as

f(x)

. If

f(x)=2x+3

, then an input of

4

gives the output

f(4)=11

.

The domain is the set of inputs for which a function is defined. For a polynomial such as

f(x)=2x+3

, every real number is allowed. For a fraction, an input that makes the denominator zero is not allowed. For example,

h(x)=\frac{1}{x-2}

is undefined at

x=2

.

When combining functions, use the same input in both functions. Then combine the outputs as directed. The allowed inputs for the result must work in both original functions. A quotient has one extra restriction: the output of the function in the denominator cannot be zero.

f(x)=2x+3\;\Rightarrow\;f(4)=11

A function rule connects an input to an output.
Check the domains before combining functions.
A quotient cannot use an input that makes its denominator function equal to zero.

2. Four ways to combine functions

Suppose

f

and

g

are functions. Their sum adds the two outputs for the same input. Their difference subtracts the output of

g

from the output of

f

. Their product multiplies the outputs. Their quotient divides the output of

f

by the output of

g

, where the output of

g

is not zero.

The order matters for subtraction and division. In general, subtracting

g

from

f

is not the same as subtracting

f

from

g

. Likewise, dividing

f

by

g

is not the same as dividing

g

by

f

.

After writing the combined rule, simplify it when possible. Keep track of restrictions from both original functions. For a quotient, also find any inputs that make the denominator function zero. Do not cancel a factor and then forget an excluded input: simplification does not restore an input where the original quotient was undefined.

(f+g)(x)=f(x)+g(x),\quad (f-g)(x)=f(x)-g(x),\quad (fg)(x)=f(x)g(x),\quad \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}

The combined functions are written $(f+g)(x)$ , $(f-g)(x)$ , $(fg)(x)$ , and $\left(\frac{f}{g}\right)(x)$ .
Use the same input in each function before combining their outputs.
For a quotient, require $g(x)\ne 0$ as well as requiring both functions to be defined.

3. Read the operations in context

The meaning of a combined function depends on what the original outputs represent. If

f(x)

is the cost of one item and

g(x)

is the cost of another item, then

(f+g)(x)

is their total cost. If the functions represent amounts added to and removed from a container, their difference can represent the change in amount.

A product can describe a total built from two changing quantities. For example, if

f(x)

is a length and

g(x)

is a width, their product can represent the area of a rectangle. A quotient can compare two quantities or describe a rate per unit. The denominator must be nonzero for that quotient to make sense.

A table can help distinguish the operation from the function rule. Each row shows how the outputs are combined for one shared input. The algebraic rules are the same whether the function values come from a formula, a table, or a graph.

Addition can represent a total, and subtraction can represent a difference.
Multiplication and division combine outputs in the same way as ordinary numbers.
Interpret the result using the meanings and units of the original functions.

4. Guided example: form and analyze all four combinations

Let

f(x)=x+2

and

g(x)=x-1

. Both rules are defined for every real input. We will write each combined rule, evaluate it at

x=3

, and identify its domain.

For the sum, add the expressions. For the difference, subtract the entire expression for

g

; brackets help preserve the minus sign. For the product, multiply the expressions. For the quotient, place

f(x)

over

g(x)

and exclude any input that makes the denominator zero.

At

x=3

, the original outputs are

f(3)=5

and

g(3)=2

. The results below can be checked by combining these two numbers. For the quotient, check the domain before evaluating:

g(x)=0

when

x=1

, so that input must be excluded.

The difference requires care with the negative sign.
The quotient has a restricted domain even though both original functions are defined for all real inputs.
Evaluating the original functions provides a numerical check on the combined rules.

5. Use the rules carefully

A reliable method is to identify the requested operation, substitute both function rules using the same variable, simplify, and then state the domain. For a quotient, solve

g(x)=0

and exclude those inputs. Also check whether either original function has its own domain restriction.

In an application, state what the combined output represents. A numerical value alone may not tell the reader whether it is a total, a difference, a product, or a comparison. Keep the units consistent with the operation and context.

The quick check asks you to apply the rules and domain restrictions. For each question, choose one answer. Use the definitions in this lesson, and remember that the order in subtraction and division is important.

Write the combined rule before evaluating it.
State any excluded inputs as part of the domain.
Check results numerically when function values are available.

How the outputs are combined

Operation	Combined output at input $x$	Domain reminder
Sum	$f(x)+g(x)$	Both functions must be defined.
Difference	$f(x)-g(x)$	Both functions must be defined; keep the order.
Product	$f(x)g(x)$	Both functions must be defined.
Quotient	$\frac{f(x)}{g(x)}$	Both must be defined, and $g(x)\ne0$ .

Worked example

Combine two linear functions

Given

f(x)=x+2

and

g(x)=x-1

, find the sum, difference, product, and quotient. State the domain of each, then evaluate each combined function at

x=3

.

Set up the sum
Add the outputs of the two functions for the same input. Simplify the resulting expression. Both original rules allow every real input, so the sum does too.
$(f+g)(x)=(x+2)+(x-1)=2x+1$
Set up the difference
Subtract the full rule for $g$ from the rule for $f$ . The brackets show that both terms in $g(x)$ are being subtracted.
$(f-g)(x)=(x+2)-(x-1)=3$
Set up the product
Multiply the two output expressions. The product remains defined for every real input because both original functions are defined for every real input.
$(fg)(x)=(x+2)(x-1)=x^2+x-2$
Set up the quotient and domain
Divide $f(x)$ by $g(x)$ . The denominator cannot equal zero. Solving $x-1=0$ gives the excluded input $x=1$ .
$\left(\frac{f}{g}\right)(x)=\frac{x+2}{x-1},\quad x\ne 1$
Evaluate and check
At $x=3$ , the original outputs are $5$ and $2$ . Add, subtract, multiply, or divide those outputs to check the four values from the combined rules.
$(f+g)(3)=7,\quad (f-g)(3)=3,\quad (fg)(3)=10,\quad \left(\frac{f}{g}\right)(3)=\frac{5}{2}$

Answer: The sum is

2x+1

, the difference is

3

, the product is

x^2+x-2

, and the quotient is

\frac{x+2}{x-1}

with

x\ne1

. Their values at

x=3

are

7

,

3

,

10

, and

\frac{5}{2}

, respectively.

Check: Since

f(3)=5

and

g(3)=2

, the four results must be

5+2=7

,

5-2=3

,

5\cdot2=10

, and

\frac{5}{2}

. The quotient is defined at

3

because its denominator value is

2

, not zero.

Common mistakes and how to avoid them

Subtracting only the first term of the second function.

Correction: Use brackets around the entire second expression. For example,

(x+2)-(x-1)

changes the sign of both terms in the second bracket.

Treating a quotient like an ordinary fraction without checking its denominator.

Correction: Find where the denominator function equals zero and exclude those inputs from the quotient's domain.

Assuming that a simplified rule automatically has the same domain as the original quotient.

Correction: Keep all restrictions from the original functions. An excluded input remains excluded even if a factor cancels during simplification.

Reversing the order of the functions in a difference or quotient.

Correction: Follow the notation exactly:

f-g

means subtract

g

from

f

, and

\frac{f}{g}

means divide

f

by

g

.

Lesson summary

Combine function outputs at the same input to form a sum, difference, product, or quotient.
Simplify the resulting rule and state its domain.
For a quotient, exclude inputs that make the denominator function zero.
Use the context to explain what the combined output represents.

Check your understanding

Question 1

If

f(x)=2x

and

g(x)=x+3

, what is

(f-g)(x)

?

$x-3$
$3-x$
$3x+3$
$\frac{2x}{x+3}$

Show answer and explanation

x-3

Subtract the full expression for

g

:

2x-(x+3)=x-3

.

Question 2

Let

p(x)=x+4

and

q(x)=x-2

. Which input must be excluded from

\left(\frac{p}{q}\right)(x)

?

$-4$
$0$
$2$
$4$

Show answer and explanation

2

The denominator function is zero when

x-2=0

, so

x=2

is not in the quotient's domain.

Question 3

If

a(5)=6

and

b(5)=-2

, what is

(ab)(5)

?

$4$
$-12$
$-3$
$8$

Show answer and explanation

-12

A product of functions multiplies their outputs at the same input:

6(-2)=-12

.

Key terms

Input: A value supplied to a function.
Output: The value returned by a function for an allowed input.
Domain: The set of inputs for which a function is defined.
Combined function: A function formed by adding, subtracting, multiplying, or dividing the outputs of two functions.

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D2.1 · Analyze sums, differences, products, and quotients of functions

Ontario Grade 12 Mathematics

Sums, differences, products, and quotients

What you will learn

1. Prerequisite bridge: inputs, outputs, and domains

2. Four ways to combine functions

3. Read the operations in context

4. Guided example: form and analyze all four combinations

5. Use the rules carefully

How the outputs are combined

Combine two linear functions

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