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Math 35 (Spring ’09) 9.1 "Algebra and Composition of Functions" Objectives: * Add, subtract, multiply, and divide functions * Find the composition of functions * Use graphs to evaluate functions Add, Subtract, Multiply, and Divide Functions Operations on Functions: If the domains and ranges of functions f and g are subsets of the real numbers, then: [Sum] [Di¤erence] [Product] [Quotient g (x) 6= 0 ] The domain of each of these functions is the set of real numbers x that are in the domain of both f and g. Example 1: (Sum/Di¤erence/Product/Quotient) Let f (x) = 4x2 9 and g (x) = 2x 3. Find the following and simplify: a) f +g b) f g c) f g d) f g Find the Composition of Functions We have seen that a function can be represented by a machine: We put in a number from the domain, and a number from the range comes out. For example: Page: 1 Notes by Bibiana Lopez Intermediate Algebra by Tussy and Gustafson 9.1 Suppose that y = f (x ) and y = g (x) de…ne two functions. Any number x in the domain of g will produce the corresponding value g (x) in the range of g. If g (x) is in the domain of function f , then g (x) can be substituted into f , and a corresponding value f (g (x)) will be determined. This two-step process de…nes a new function called a , denoted by ("f composed with g" or "the composition of f and g"). The function machines can illustrate the composition f g. When we put a number into the function g, a value g (x) comes out. The value g (x) then goes into function f , which transforms g (x) into f (g (x)) ("f of g of x"). If the function machines for g and f were connected to make a single machine, that machine would be named f g: Composite Functions: The composite function f g is de…ned by: Example 2: (Composite functions) Let f (x) = 2x + 1 and g (x) = x a) (f c) (g f ) ( 2) 4: Find: g) (9) Page: 2 b) (f g) (x) d) (f g) ( 2) Notes by Bibiana Lopez Intermediate Algebra by Tussy and Gustafson 9.1 Example 3: (Composite functions) Let f (x) = 3x a) (f g) (2) c) (f g) (x) 2 and g (x) = x2 + x + 1: Find: b) d) (g f ) (2) (g f ) (x) Example 4: (Composite functions) If f (x) = x + 1 and g (x) = 2x 5; show that (f g) (x) 6= (g f ) (x) Use Graphs to Evaluate Functions Example 6: (Using graphs to evaluate functions) Refer to the graphs of functions f (red) and g (blue) to …nd each of the following a) (f + g) ( 4) b) (f g) (2) c) (f d) (f g) (3) f ( 2) g e) f) y 4 2 g) ( 3) (g f ) (3) 6 -6 -4 -2 2 -2 4 6 x -4 -6 Page: 3 Notes by Bibiana Lopez