Domain of Combined Functions

Pre-Calculus · Axiom Academy

LESSON Domain of Combined Functions When you add, subtract, multiply, or divide two functions, where is the result actually defined? The answer is where their domains overlap. For f+g , f-g , and , every input must be fed into both f and g . So the combined function is defined exactly where the domain of f and the domain of g overlap — their intersection. Both functions must be defined at x 2. The Rule for Each Operation Three of the four operations share the exact same domain. Division is the odd one out, because dividing by zero is undefined — so we must also throw out any x where the denominator g(x)=0 . For every operation, begin with — the inputs both functions accept. For f/g , remove each x with g(x)=0 . The point leaves the interval as an open gap. Find the domain of (f+g) and for the functions below. These two operations share a domain, so one answer covers both. Now the division case, where the extra restriction actually does something. Here both domains are bounded, so the intersection is a real interval — and then we punch out the zero of g . f+g , f-g , would all have domain [-2,4] — the plain intersection. f/g loses the single point x=4 , leaving [-2,4) . One hole, real consequence. The domain of a combined function is the overlap of the two domains — and division alone removes the inputs that make the denominator zero. Scroll up to revisit any step.

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