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 $f + g$	$(f + g) (x) = f (x) + g (x)$	$Dom (f) \cap Dom (g)$
Difference $f - g$	$(f - g) (x) = f (x) - g (x)$	$Dom (f) \cap Dom (g)$
Product $f \cdot g$	$(f \cdot g) (x) = f (x) \cdot g (x)$	$Dom (f) \cap Dom (g)$
Quotient $f / g$	$(f / g) (x) = f (x) / g (x)$	$Dom (f) \cap Dom (g) \cap {x ∣ g (x) \neq 0}$

Arithmetic of Functions

Given two functions $f$ and $g$ , we define:

The domains of $f + g$ , $f - g$ , and $f \cdot g$ are

Dom (f) \cap Dom (g),

the set of $x$ -values where both $f (x)$ and $g (x)$ are defined simultaneously.

For $f / g$ , we additionally exclude any $x$ where $g (x) = 0$ :

Dom (f / g) = Dom (f) \cap Dom (g) \cap {x ∣ g (x) \neq 0} .

Let $f (x) = \frac{1}{x - 3}$ and $g (x) = \sqrt{x - 2}$ . Find $f + g$ , $f - g$ , $f \cdot g$ , $f / g$ , and $g / f$ , with their domains. Also evaluate each at $x = 5$ .

Solution

Domains of $f$ and $g$ :

Dom (f) = ℝ - {3}, Dom (g) = [2, \infty) .

Intersection:

{x ∣ x \geq 2 and x \neq 3} = [2, 3) \cup (3, \infty)

. Combined functions and their domains:

(f + g) (x) = \frac{1}{x - 3} + \sqrt{x - 2}, Dom = [2, 3) \cup (3, \infty) .

(f - g) (x) = \frac{1}{x - 3} - \sqrt{x - 2}, Dom = [2, 3) \cup (3, \infty) .

(f \cdot g) (x) = \frac{\sqrt{x - 2}}{x - 3}, Dom = [2, 3) \cup (3, \infty) .

For

f / g

, also exclude

x = 2

where

g (2) = 0

(\frac{f}{g}) (x) = \frac{1}{(x - 3) \sqrt{x - 2}}, Dom = (2, 3) \cup (3, \infty) .

Since

f (x) = 1 / (x - 3) \neq 0

for all

x

in its domain:

(\frac{g}{f}) (x) = (x - 3) \sqrt{x - 2}, Dom = [2, 3) \cup (3, \infty) .

Values at $x = 5$ :

A Subtle Domain Issue with Roots

Are the functions $h_{1} (x) = \sqrt{x - 1} \sqrt{2 - x}$ and $h_{2} (x) = \sqrt{(x - 1) (2 - x)}$ equal? What about $u_{1} (x) = \sqrt{x - 1} \sqrt{x - 2}$ and $u_{2} (x) = \sqrt{(x - 1) (x - 2)}$ ?

Solution

(a) $h_{1}$ vs. $h_{2}$ :

Dom (h_{1}) = [1, \infty) \cap (- \infty, 2] = [1, 2]

. For

h_{2}

, we need

(x - 1) (2 - x) \geq 0

. A sign chart shows this holds on $[1,2]$. So

Dom (h_{1}) = Dom (h_{2}) = [1, 2]

, and

h_{1} (x) = h_{2} (x)

on $[1,2]$. They are equal.

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

(b) $u_{1}$ vs. $u_{2}$ :

Dom (u_{1}) = [1, \infty) \cap [2, \infty) = [2, \infty)

. For

u_{2}

, we need

(x - 1) (x - 2) \geq 0

. A sign chart shows this holds on

(- \infty, 1] \cup [2, \infty)

. So

Dom (u_{1}) = [2, \infty)

but

Dom (u_{2}) = (- \infty, 1] \cup [2, \infty)

. The domains differ, so

u_{1}

and

u_{2}

are not equal.

Sign chart for (x-1)(x-2) with roots at x=1 and x=2. — Sign chart for $(x - 1) (x - 2)$ : product is nonnegative on $(- \infty, 1] \cup [2, \infty)$ .

Graphs of Combined Functions

To graph $f + g$ : for each $x$ , add the $y$ -coordinates of $f$ and $g$ geometrically.

The graphs of $f (x) = x$ and $g (x) = x^{3} - 4 x + 1$ are given. Sketch the graph of $h = f + g$ by graphical addition.

Solution

h (x) = f (x) + g (x) = x^{3} - 3 x + 1

. For each

x

, the height of

h

equals the sum of the heights of

f

and

g

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

At the points where

f

and

g

intersect (approximately

x \approx - 2.3, 0.2, 2.1

h (x) = 2 f (x) = 2 g (x)

Resulting graph of h(x) = x³ - 3x + 1 by graphical addition. — Graph of $h (x) = x^{3} - 3 x + 1$ 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

x

is in both

Dom (f)

and

Dom (g)

, if

g (x) = 0

at that point, the quotient

f (x) / g (x)

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

f / g

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

No. The domain of

f + g

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

f

and

g

means: at each

x

-value in the common domain, stack the

y

-value of

f

on top of the

y

-value of

g

to get the

y

-value of

f + g

. This can be done visually with a ruler.

\sqrt{a} \cdot \sqrt{b}

always equal to

\sqrt{a b}

Only when both

a \geq 0

and

b \geq 0

. If one is negative,

\sqrt{a}

\sqrt{b}

is not real, but

\sqrt{a b}

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

\sqrt{f} \cdot \sqrt{g}

and

\sqrt{f g}

can differ.

Quick Reference

Arithmetic of Functions

A Subtle Domain Issue with Roots

Graphs of Combined Functions

Frequently Asked Questions

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