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1.4 BUILDING FUNCTIONS FROM FUNCTIONS By: Alaina Riedel & Lauren Evenson “Building Blocks, Building Functions” g(x) = f(x) = f(x) = 2 f(x) = + 4 http://www.youtube.com/watch?v=9rsJF6lqxao Combining Functions A way of creating new functions to combine two or more functions to create a new function. The most obvious way we can do this is to perform basic algebraic operations on the two functions to create the new one Add, subtract, multiply or divide functions. The algebra of real numbers: 4 x 5 = 20, 4 - 5 = -1 The algebra of functions: fg, f-g, etc Sum, Difference, Product, and Quotient Composition of Functions Relations and Implicitly Defined Functions - Relation: A set of ordered pairs of real numbers If the relation happens to relate a single value of y to each value of x, then the relation is also a function and its graph will pass the vertical line test Implicitly defined function: A function that is a subset of a relation defined by an equation in x and y Example - Relation Determine which of the ordered pairs (2,-5), (1,3), and (2,1) are in the relation defined by 2 y + 2 = 5. Is this relation a function? (2,-5): (2)2 (5) + (-5)2 = 5 (1,3): (1)2 (3) + (3)2 = 12 = 5 (2,1): (2)2 (1) + (1)2 = 5 So, (2,-5) and (2,1) are in the relation, but (1,3) is not. Since the equation relates two different y-values (-5 and 1) to the same x-value (2), the relation cannot be a function. Example - Implicitly The graph consists of two parallel lines, each the graph of one of the implicitly defined functions Alaina & Lauren Mix Up jeopardylabs.com /play/building-functions-from-functions2 Materials Used Pre-Calculus Book http://facultypages.morris.umn.edu/~mcquarrb/te achingarchive/Precalculus/Lectures/BuildingFunction fromFunctions.pdf http://math.ucsd.edu/~wgarner/math4c/textbook/ chapter2/combfunctions.htm Jeopardy Game: https://jeopardylabs.com/