FUNCTION FACTS, Lecture notes of Algebra

Download FUNCTION FACTS and more Lecture notes Algebra in PDF only on Docsity! FUNCTION FACTS Definition of a Function: A function is a rule that describes how one quantity depends upon another. • 𝑓𝑓(𝑥𝑥) = 𝑦𝑦 is read “𝑓𝑓 of x.” • The output variable, 𝑦𝑦 is the dependent variable because it depends on the input variable, x which is called the independent variable. • For each input x, there is only one possible output y. Example: The set of points {(1,2), (2,4), (3,-1), (4,4)} is a function. The set of points {(1,2), (2,4), (3,-1), (3,4)} is not a function since an input of 3 yields more than one output. Vertical Line Test: This tests whether or not a relation between two variables is a function. If a vertical line crosses the curve more than once, the relation is not a function. Domain: The domain is the set of all possible values of x for which the function 𝑓𝑓(𝑥𝑥) exists. • x cannot cause a denominator to be zero. • If x is under a square root (or any even root) sign, x cannot cause the expression under the root sign to be negative (when using real numbers). • x must be greater than 0 for 𝑦𝑦 = log𝑏𝑏 𝑥𝑥 . Range: The range is the set of all possible values of the function, that is, the output variable, 𝑦𝑦. Values of Functions: Example: Let 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2 + 4𝑥𝑥 − 3 . Find 𝑓𝑓(2): 𝑓𝑓(2) = 22 + 4(2) − 3 = 4 + 8 − 3 = 9 Find 𝑓𝑓(𝑥𝑥 + 1): 𝑓𝑓(𝑥𝑥 + 1) = (𝑥𝑥 + 1)2 + 4(𝑥𝑥 + 1) − 3 = 𝑥𝑥2 + 6𝑥𝑥 + 2 Algebra of Functions: Sum: (𝑓𝑓 + 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥) Difference: (𝑓𝑓 − 𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) − 𝑔𝑔(𝑥𝑥) Product: (𝑓𝑓𝑔𝑔)(𝑥𝑥) = 𝑓𝑓(𝑥𝑥) ∙ 𝑔𝑔(𝑥𝑥) Quotient: �𝑓𝑓 𝑔𝑔 � (𝑥𝑥) = 𝑓𝑓(𝑥𝑥) 𝑔𝑔(𝑥𝑥) for 𝑔𝑔(𝑥𝑥) ≠ 0 This is a function. This is not a function.