Domain Rules and Examples for Combined Functions (copy)

Domain of combined functions

The domain of a new function formed by combining two functions f and g is determined by the intersection of the domains of f and g.

Domain of f + g

The domain of the sum f + g is given by Dom(f) ∩ Dom(g), which includes all x values that satisfy both domains.

Additional restriction for division

In the case of f ÷ g, the additional restriction is that g(x) cannot be zero.

Symbol for empty intersection

The notation for the empty set, representing no common values in the domains of f and g, is often denoted by ∅.

Square root function domain restriction

For a square root function, the radicand must be nonnegative; hence, the argument must satisfy the condition 1 - x ≥ 0.

Rational function domain restriction

For a rational function, the denominator cannot be zero, which means you must solve the denominator equation to find excluded values.

Domain of f ÷ g

To find the domain of f ÷ g, verify Dom(f) ∩ Dom(g) and ensure g(x) ≠ 0 for all x in the intersection.

Polynomial functions domain

The domain of polynomial functions is all real numbers, expressed as ℝ.

Linear function domain

The domain of a linear function is also all real numbers, but can have a specific excluded value where the function equals zero.

Holes in one function from cancellation

If factors cancel during simplification, check the original domain to ensure no points of discontinuity (holes) are ignored.