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Onward to Section 1.4a… Combining functions algebraically, composite functions, and decomposing functions! Definition: Sum, Difference, Product, and Quotient of Functions Let f and g be two functions with intersecting domains. Then for all values of x in the intersection, the algebraic combinations of f and g are defined by the following rules: Sum: Difference: Product: Quotient: f g x f x g x f g x f x g x fg x f x g x f x f , provided g x 0 x g x g In each case, the domain of the new function consists of all numbers that belong to both the domain of f and the domain of g. As noted, the zeros of the denominator are excluded from the domain of the quotient. Guided Practice For the given functions, find f + g, f – g, fg, f/g, and gg. Give the domain of each. f x x 2 g x x 1 First, find the domain of the original functions, and determine where these two domains intersect (overlap). Domain of f: , Domain of g: Domain intersection: 1, 1, This intersection becomes the domain of all of the algebraic combination functions!!! Guided Practice For the given functions, find f + g, f – g, fg, f/g, and gg. Give the domain of each. f x x 2 g x x 1 f g x f x g x x x 1 2 with D: 1, f g x f x g x x x 1 2 with D: 1, fg x f x g x x x 1 with D: 1, 2 Guided Practice For the given functions, find f + g, f – g, fg, f/g, and gg. Give the domain of each. f x x 2 f x f x g x g g x x 1 2 x x 1 gg x g x g x x 1 with D: 1, 2 with D: 1, Can we simplify this last one??? Guided Practice For the given functions, find formulas for the functions f + g, f – g, and fg. Give the domain of all functions. f x x 5 D : 5, g x x 3 D : , f g x x 5 x 3 f g x x 5 x 3 fg x x 3 x 5 Domain of all 3 combination functions: 5, Guided Practice For the given functions, find formulas for f/g and g/f. Give the domain of all functions. f x x 2 D : 2, g x x 4 D : 4, f x g x2 x4 x2 D : 2, x4 g x f x4 x2 x4 D : 2, x2 Now on to composite functions?! Definition: Composition of Functions Let f and g be two functions such that the domain of f intersects the range of g. The composition of f and g, denoted f g, is defined by the rule f g x f g x The domain of f g consists of all x-values in the domain of g that map to g(x)-values in the domain of f. NOTE: In most cases, f g and g f are different functions!!! A Few Practice Problems… For the given functions, find (f g)(x) and (g f)(x) and verify (both algebraically and graphically) that the two composite functions are not the same. f x ex f g g x x x e f x g f x g e e g x f g x f x Now, how do we verify??? x x A Few Practice Problems… For the given functions, find (f g)(x) and (g f)(x) and give the domain of each composition function. f x x2 1 g x x f g x g f x x x 1 2 2 1 D: D: 0, , 1 1, Let’s check these with the calculator… A Few Practice Problems… For the given functions, find (f g)(3) and (g f)(–2). f x x 1 g x 2 x 3 2 f g 3 f g 3 8 g f 2 g f 2 3 Decomposing Functions…working backwards or undoing a composition… For each function h, find functions f and g such that h(x) = f(g(x)). h x x 1 3 x 1 4 2 f x x 3x 4 2 g x x 1 Decomposing Functions For each function h, find functions f and g such that h(x) = f(g(x)). h x x 1 3 f x x g x x 1 3 Any other ways to solve this one?!?! Let’s do some modeling… In math-land, not fashion-land… In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second. 1. Find the volume after t seconds. V = 44t Let’s do some modeling… In math-land, not fashion-land… In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second. 2. When the volume is V, what is the radius r ? 4 3 πr V 3 3V r 4π 3 3 V r3 4π Let’s do some modeling… In math-land, not fashion-land… In the medical procedure known as angioplasty, doctors insert a catheter into a heart vein and inflate a small, spherical balloon on the tip of the catheter. Suppose the balloon is inflated at a constant rate of 44 cubic millimeters per second. 3. Write an equation that gives the radius r as a function of the time. What is the radius after 5 seconds? V 44t and 3 V r3 4π 3 44 t 33 t 3 r 3 4π π At 5 seconds, r = 3.745 mm Whiteboard Practice For the given functions, find formulas for the functions f + g, f – g, and fg. Give the domain of all functions. f x x 1 2 g x 3 x 2 x 3x 4 f g x 2 x x2 f g x fg ( x ) x 3 5 x 2 7 x 3 Domain of all five functions: , Whiteboard Practice Find f(g(x)) and g(f(x)). State the domain of each. 1 2 g ( x) f ( x) x 1 x 1 1 f ( g ( x )) 1 2 ( x 1) D : ( ,1) (1, ) 1 x2 2 D : ( , 2) ( 2, 2) ( 2, ) g ( f ( x )) Whiteboard Practice Find f(x) and g(x) so that the function can be described as y=f(g(x)). y ( x 1) 3 2 f ( x ) ( x 1)2 g ( x) x 3 Homework: p. 127-128 1-23 odd one possible solution…