Algebraic Combinations of Functions

Just as we combine numbers with arithmetic, we can combine functions by addition, subtraction, multiplication, and division. The resulting function's domain is determined by where all component functions are simultaneously defined.

Quick Reference

Operation	Formula	Domain
Sum
Difference
Product
Quotient

Arithmetic of Functions

Given two functions and , we define:

The domains of , , and are

the set of -values where both and are defined simultaneously.

For , we additionally exclude any where :

Let and . Find , , , , and , with their domains. Also evaluate each at .

Solution

Domains of and :

Intersection:

. Combined functions and their domains:

For

, also exclude

where

Since

for all

in its domain:

Values at :

A Subtle Domain Issue with Roots

Are the functions and equal? What about and ?

Solution

(a) vs. :

. For

, we need

. A sign chart shows this holds on

. So

, and

. They are equal.

Sign chart for (x-1)(2-x) with roots at x=1 and x=2. — Sign chart for : product is nonnegative on .

(b) vs. :

. For

, we need

. A sign chart shows this holds on

. So

but

. The domains differ, so

and

are not equal.

Sign chart for (x-1)(x-2) with roots at x=1 and x=2. — Sign chart for : product is nonnegative on .

Graphs of Combined Functions

To graph : for each , add the -coordinates of and geometrically.

The graphs of and are given. Sketch the graph of by graphical addition.

Solution

. For each

, the height of

equals the sum of the heights of

and

Graphs of f(x) = x and g(x) = x³ - 4x + 1. — Graphs of and .

At the points where

and

intersect (approximately

Resulting graph of h(x) = x³ - 3x + 1 by graphical addition. — Graph of obtained by graphical addition.

Frequently Asked Questions

Why does the domain of f/g exclude points where g(x) = 0?

Division by zero is undefined. Even if

is in both

and

, if

at that point, the quotient

does not exist. We must exclude those points from the domain of

Can the domain of f + g be larger than the domain of either f or g?

No. The domain of

is a subset of each individual domain. It is the intersection, which is at most as large as either factor and often smaller.

What does graphical addition mean?

Graphical addition of

and

means: at each

-value in the common domain, stack the

-value of

on top of the

-value of

to get the

-value of

. This can be done visually with a ruler.

always equal to

Only when both

and

. If one is negative,

is not real, but

might be (when both are negative, their product is positive). This is why the domains of

and

can differ.

Quick Reference

Arithmetic of Functions

A Subtle Domain Issue with Roots

Graphs of Combined Functions

Frequently Asked Questions

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