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Sum: 
The sum of two functions is a function. Suppose you have
two functions f and g. A function
h, defined as $h(x)=f(x)+g(x)$, is called the sum
of f and g. Every value of h is a
sum of a value of f and a value of g.
The sum function is conventionally written as "$(f+g)$". So $(f+g)(x)=f(x)+g(x)$ Take note: The character "+" in "$(f+g)$" has a diffent meaning than "+" in $f(x)+g(x)$. In "$(f + g)$", "+" is a character in the sum function's name. In "$f(x)+g(x)$", "+" is the arithmetical operation "plus". A sum of two functions is defined only for values where each function is defined. For example, let f be defined as $f(x)=2x$ and let g be defined as $g(x) = 1/(x2)$. Then $(f+g)(2)$ is undefined because $g(2)$ is undefined. 


Product: 
The product of two functions is a function. Suppose you
have two functions f and g. A function
p, defined as $p(x)=f(x)\cdot g(x)$, is called
the product of f and g. Every value of
p is a product of a value of f and a value
of g.
The product function is conventionally written as "$(f\times g)$" or, more commonly, as "$(f \cdot g)$". So $(f\cdot g)(x)=f(x)\cdot g(x)$ Take note: The character "$\cdot$" in "$(f\cdot g)$" has a diffent meaning than "$\cdot$" in $f(x)\cdot g(x)$. In "$(f\cdot g)$", "$\cdot$" is a character in the product function's name. In "$f(x)\cdot g(x)$", "$\cdot$" is the arithmetical operation "times". A product of two functions is defined only for values where each function is defined. For example, let f be defined as $f(x)=2x$ and let g be defined as $g(x)=1/(x2)$. Then $(f\cdot g)(2)$ is undefined because $g(2)$ is undefined. 


Quotient: 
The quotient of two functions is a function. Suppose you
have two functions f and g. A function
q, defined as $q(x)= f(x)/g(x)$, is called the
quotient of f and g. Every value of
q is a quotient of a value of f and a
value of g. If f and g are
polynomial functions, then q is called a rational
function.
The quotient function is conventionally written as "$(f / g)$". So $(f / g)(x)=f(x)/g(x)$ Take note: The character "/" in "$(f/g)$" has a diffent meaning than "/" in $f(x)/g(x)$. In "$(f/g)$", "/" is a character in the quotient function's name. In "$f(x)/g(x)$", "/" is the arithmetical operation "division". A quotient of two functions f and g is defined only for values where each function is defined and for values of x such that $g(x)\neq 0$. For example, let f be defined as f(x) = 2x and let g be defined as $g(x)=2x$. Then $(f/g)(2)$ is undefined because $g(2)=0$. 


Power: 
A power of a function is a function. Suppose you have
two functions f and g. A function
s, defined as $s(x)=f(x)^{g(x)}$, is called
f to the power of g. Every value of
s is a value of f raised to a power of
g. If f is a constant function, such as
$f(x)=3$, then
$s(x)=3^{g(x)}$ is called an exponential function.
The power function is conventionally written as "$(f \,\text{^}\, g)$". So $(f \,\text{^}\, g)(x)=f(x)^{g(x)}$ Take note: The character "^" in "$(f\,\text{^}\, g)$" has a diffent meaning than "^" in $f(x)\,\text{^}\,g(x)$ (or $f(x)^{g(x)}$). In "$(f\,\text{^}\,g)$", "^" is a character in the power function's name. In "$f(x)\,\text{^}\,g(x)$", "^" is the arithmetical operation "power". A power function is defined only for values where each function is defined. Also, a power of f is undefined when $f(x)\lt 0$ and $g(x)$ is not an integer. For example, let f be defined as $f(x)=2x$ and let g be defined as $g(x)=1/(x2)$. Then $(f\,\text{^}\,g)(2)$ is undefined because $g(2)$ is undefined. Similarly, if $f(x)=\cos(x)$ and $g(x)=\sin(x)$, then $g(x)\,\text{^}\,f(x)$ is undefined whenever $\sin(x)\lt 0$. 


Composite: 
Suppose you have two functions f and g. A
function c, defined as $c(x)=f(g(x))$
is called the
composition of f and g. Every value of
c is the value of f evaluated at a value
of g.
You have seen many examples of composite functions. For example, when we define c as $c(x)= \sin(3x+2)$ we have defined a composite function. Think of g as $g(x)=3x+2$ and think of f as $f(x)=\sin(x)$. Then $f(g(x))=\sin(3x+2)$. Any function with a complicated argument is a composite function. The composite function is conventionally written as "$(f \circ g)$". So $(f \circ g)(x)=f(g(x))$. The composite function $f(g(x))$ is defined only for values of f that are in the domain of g’s independent variable. 
The case of a composite function deserves special clarification. In the description of the composite function c, we said "Every value of c is the value of g evaluated at a value of f." The animation in Figure 3.16.1 is designed to clarify this statement. The animations illustrates one way to think of evaluating $g(f(x))$ at $x = a$. It shows a being assigned as a generic value of x, then evaluating f at $x = a$ to get $f(a)$. Then the value of $f(a)$ is assigned as the value of u, g’s independent variable, getting the value $g(f(a))$ as the value of g at $u = f(a)$. Finally, a, as a value of x, is paired with the value $g(f(a))$.
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