Let $f$ and $g$ be two real functions given by $f = \{(0, 1), (2, 0), (3, -4), (4, 2), (5, 1)\}$, $g = \{(1, 0), (2, 2), (3, -1), (4, 4), (5, 3)\}$ then the domain of $f \cdot g$ is given by _________.

Let f and g be two real functions given by f = \(0, 1), (2, 0), (3, -4), (4, 2), (5, 1)\, g = \(1, 0), (2, 2),

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Concept understanding — Function Operations

Function Operations: Combining Machines

Think of a function as a machine. You feed it an input (say, a number $x$ ), it does something, and out comes an output $f(x)$ . Now imagine you have two such machines, $f$ and $g$ . Function operations are simply ways to hook these machines together — to add, subtract, multiply, or divide their outputs, or to feed one machine's output into the other.

The core idea is simple: if you can do arithmetic with numbers, you can do arithmetic with functions. The only catch is that both functions must be "ready to work" on the same input at the same time.

The Four Arithmetic Operations

Let $f$ and $g$ be two functions. For any input $x$ that belongs to both their domains (the set of numbers each can accept), we define:

Operation	Notation	What it means
Sum	$(f+g)(x)$	$f(x) + g(x)$
Difference	$(f-g)(x)$	$f(x) - g(x)$
Product	$(f \cdot g)(x)$	$f(x) \cdot g(x)$
Quotient	$\left(\frac{f}{g}\right)(x)$	$\frac{f(x)}{g(x)}$ , provided $g(x) \neq 0$

Note

The domain of the new function is the intersection of the domains of $f$ and $g$ — the numbers both machines can handle. For the quotient, you must also exclude any $x$ where $g(x) = 0$ , because division by zero is undefined.

Example. Let $f(x) = \sqrt{x}$ (domain: $x \geq 0$ ) and $g(x) = x - 1$ (domain: all real numbers). Then:

$(f+g)(x) = \sqrt{x} + x - 1$ , domain: $x \geq 0$ .
$\left(\frac{f}{g}\right)(x) = \frac{\sqrt{x}}{x-1}$ , domain: $x \geq 0$ and $x \neq 1$ .

Composition: Feeding One Machine into Another

This is the most powerful operation. Instead of adding outputs side by side, you take the output of one function and feed it as the input to the other.

(f \circ g)(x) = f\big(g(x)\big)

Read " $f$ composed with $g$ ". You do $g$ first, then $f$ on the result.

Intuition. Suppose $g$ is a machine that converts Celsius to Fahrenheit, and $f$ is a machine that converts Fahrenheit to Kelvin. Then $f \circ g$ converts Celsius directly to Kelvin — one combined machine.

Domain trap. For $f(g(x))$ to make sense, two conditions must hold:

$x$ must be in the domain of $g$ (so $g(x)$ exists).
$g(x)$ must be in the domain of $f$ (so $f$ can accept it).

So the domain of $f \circ g$ is: all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$ .

Watch out

Composition is not commutative. $f \circ g$ is almost never the same as $g \circ f$ . For example, if $f(x) = x^2$ and $g(x) = x+1$ , then:

$(f \circ g)(x) = (x+1)^2 = x^2 + 2x + 1$
$(g \circ f)(x) = x^2 + 1$ …

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The domain of the product of two functions, $f \cdot g$ , is the set of all $x$ -values for which both $f(x)$ and $g(x)$ are defined. This means it is the intersection of the individual domains of $f$ and $g$ . For the given functions, the domain of $f \cdot g$ is $\boxed{\{2, 3, 4, 5\}}$ .

When we talk about functions like $f$ and $g$ being sets of ordered pairs, each pair $(x, y)$ means that for the input $x$ , the function gives an output $y$ . The set of all possible input values ( $x$ -coordinates) for which a function is defined is called its domain.

When we perform operations like addition, subtraction, multiplication, or division on two functions, say $f$ and $g$ , we are essentially creating a new function. For this new function to be defined at a particular input $x$ , both original functions, $f$ and $g$ , must be defined at that same input $x$ .

Consider the product function, $(f \cdot g)(x)$ . By definition, this is equal to $f(x) \cdot g(x)$ . For this product $f(x) \cdot g(x)$ to have a meaningful value, $f(x)$ must exist, and $g(x)$ must also exist. If $f(x)$ is undefined for some $x$ , or if $g(x)$ is undefined for that $x$ , then their product $f(x) \cdot g(x)$ cannot be computed.

This leads to a fundamental rule for the domain of combined functions:

f \cdot g

Let's apply this concept to the given functions.

Identify the domain of function $f$ .

The function $f$ is given as a set of ordered pairs: $f = \{(0, 1), (2, 0), (3, -4), (4, 2), (5, 1)\}$ .

The domain of $f$ , $D_f$ , consists of all the first elements (the $x$ -coordinates) of these ordered pairs.

$D_f = \{0, 2, 3, 4, 5\}$
Identify the domain of function $g$ .

The function $g$ is given as a set of ordered pairs: $g = \{(1, 0), (2, 2), (3, -1), (4, 4), (5, 3)\}$ .

The domain of $g$ , $D_g$ , consists of all the first elements (the $x$ -coordinates) of these ordered pairs.

$D_g = \{1, 2, 3, 4, 5\}$ …

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