Combining Functions Summary

Pre-Calculus · Axiom Academy

A recap of building new functions from old ones: arithmetic combinations, their domains, composition, and decomposition. You can add, subtract, multiply, and divide functions point-by-point: (f+g)(x)=f(x)+g(x) , and likewise for - , , and . The domain of an arithmetic combination is the intersection of the two domains — and for f/g you must also throw out every x with g(x)=0 . Composition feeds g 's output into f . Order matters: in general . The domain of is a two-stage gate: x must be in the domain of g , and g(x) must be in the domain of f . Decomposition runs the process backward — write a complicated h(x) as f(g(x)) by spotting an inner function and an outer function. Core Concept Arithmetic Combinations Combine two functions by doing the arithmetic on their output values at each input x — add, subtract, multiply, or divide the numbers f(x) and g(x) . When to use: building one function from two simpler pieces. Watch out for: (fg)(x) is a product, not a composition — don't plug g into f . Core Concept Domains of Combinations For f+g , f-g , and fg , an input is allowed only when both functions accept it — so the domain is the intersection. For the quotient f/g , also delete every x that makes the denominator zero. When to use: stating where a combined function is defined. Watch out for: forgetting the g(x)=0 exclusion for f/g .

This is the written version of the interactive lesson above. See the full Pre-Calculus course.