(Section 1.6: Combining Functions) 1.6.5 PART C: LINEAR COMBINATIONS OF FUNCTIONSLetfand gbe functions. Let cand dbe real numbers, possibly 0. cf+dgis then called a linear combinationoffand g. • Its domain is the overlap (intersection) Domf()±Domg. • More generally, a linear combination of objects is a sum of constant multiples of those objects. Example 4 (Linear Combinations)a) 3f+4g, 12f±g, ±g, and 0 are linear combinations offand g. b) 2f+3g±4hand ±3.7f+gare linear combinations off, g, and h. §• In Section 1.7, we will see that a linear combination of evenfunctions is even. The same goes for oddfunctions. • WARNING 4:People often erroneously attempt to apply linearity properties in precalculus. For example, remember that the “square root of a sum” is typically notequal to the “sum of the square roots.” Think: +4±+4. • In calculus, we will see important linearity theoremsfor limits, derivatives, and integrals.
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