Combining Functions

Functions can be combined to form new mathematical models that describe interactions between variables. These combinations are fundamental in understanding relationships between changing quantities and are commonly encountered in scientific and engineering contexts. The combination methods—addition, subtraction, multiplication, division, and composition—each have unique implications for the resulting function’s domain and behavior.

When combining functions through arithmetic operations, such as

The domain of the resulting function is restricted to the set of input values common to the domains of both f and g. In the division case, additional restrictions are imposed to exclude any input x for which g(x) = 0, as division by zero is undefined.

For instance, if f(x) = √x and g(x) = x - 2, then (f + g)(x) is defined for x ≥ 0 since the square root function requires non-negative inputs. However,(f/g)(x) = f(x)/g(x) is only defined for x > 2, where both f(x) is real and g(x) ≠ 0.

In function composition, the output of one function becomes the input of another, denoted as (f ⚬ g)(x) = f(g(x)). The domain of f ⚬ g includes all x such that x is in the domain of g, and g(x) lies within the domain of f. This structure models sequential processes, where one quantity depends on another, which in turn depends on a third.

The domain of a composite function must account for both inner and outer functions. For example, if f(x) = √x and g(x) = x² - 9, then (f ⚬ g)(x) = √(x² - 9)​, which requires x² - 9 ≥ 0 giving a domain of (−∞,−3]∪ [3,∞). Each step in the function must yield outputs valid for the next step in the composition.