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Sum: | The sum of two functions must be 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 values of f and g. Take note: A difference of two functions is also a sum. 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, let g be defined as $g(x) = 1/(x-2)$, and let h be defined as $h(x)=f(x) + g(x)$. Then h(2) is undefined because g(2) is undefined. |
Product: | The product of two functions must be a function. Suppose you have two functions f and g. A function p, defined as $p(x) = f(x)g(x)$, is called the product of f and g. Every value of p is a product of values of f and g. 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, let g be defined as g(x) = 1/(x-2), and let h be defined as $h(x)=f(x)g(x)$. Then h(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 $q(x) = \frac{f(x)}{g(x)}$, is called the quotient of f and g. Every value of q is a quotient of values of f and g. If f and g are polynomial functions, then q is called a rational function. 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 $g(x) ≠ 0$. For example, let f be defined as f(x) = 2x, let g be defined as g(x) = (x-2), and let h be defined as $h(x)=f(x)/g(x)$. Then h(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 the power of f and g, or f to the power of g. Every value of s is a power of values of f and g. If f is a constant function, such as $f(x) = 2$ for all values of x, then s is called an exponential function. A power function is defined only for values where each function is defined and for values of x such that $f(x) > 0$. For example, let f be defined as f(x) = 2x, let g be defined as g(x) = 1/(x-2), and let h be defined as $h(x)=f(x)^{g(x)}$. Then h(2) is undefined because g(2) is undefined. |
Composite: | Suppose you have two functions f and g. A function
c, defined as $c(x) = g(f(x))$ is called the composite of f and g, or the composition of f and g. Every value of c is the value of g evaluated at a value of f. You have seen many examples of composite functions. When we defined c as $c(x) = \sin(3x + 2)$, we defined a composite function. Think of g as g(x) = sin(x) and think of $3x + 2$ as $f(x) = 3x + 2$. Then $c(x) =g(f(x))$. Any function with a complicated argument is a composite function. The composite function g(f(x)) is defined only for values of f that are in the domain of g’s independent variable. |