Precalc0105to0107-page17 - (Section 1.6: Combining...

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(Section 1.6: Combining Functions) 1.6.5 PART C: LINEAR COMBINATIONS OF FUNCTIONS Let f and g be functions. Let c and d be real numbers, possibly 0. cf + dg is then called a linear combination of f and g . • Its domain is the overlap (intersection) Dom f () ± Dom g . • More generally, a linear combination of objects is a sum of constant multiples of those objects. Example 4 (Linear Combinations) a) 3 f + 4 g , 1 2 f ± g , ± g , and 0 are linear combinations of f and g . b) 2 f + 3 g ± 4 h and ± 3.7 f + g are linear combinations of f , g , and h . § • In Section 1.7, we will see that a linear combination of even functions is even . The same goes for odd functions. WARNING 4 : People often erroneously attempt to apply linearity properties in precalculus. For example, remember that the “square root of a sum” is typically not equal to the “sum of the square roots.” Think: + 4 ± + 4 . In calculus , we will see important linearity theorems for limits, derivatives, and integrals.
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