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DO NOW:

DO NOW:. Find the domain and range of f(x). 1.4 Building Functions From Functions. Combining Functions Algebraically:. Sum: (f +g)(x) = f(x) + g(x) Difference: (f - g)(x) = f(x) - g(x) Product: (fg)(x) = f(x)g(x) Quotient: (f/g)x = f(x)/g(x), provided g(x)≠0

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DO NOW:

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  1. DO NOW: • Find the domain and range of f(x)

  2. 1.4 Building Functions From Functions

  3. Combining Functions Algebraically: • Sum: (f +g)(x) = f(x) + g(x) • Difference: (f - g)(x) = f(x) - g(x) • Product: (fg)(x) = f(x)g(x) • Quotient: (f/g)x = f(x)/g(x), provided g(x)≠0 • ***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.

  4. Defining new functions algebraically: Let f(x) = x2 and g(x) = √(x+1) Find formulas for the functions f + g, f - g, fg, f/g, and gg. Give the domain of each.

  5. Composition of Functions: • The composition f of g, denoted f○g: (f ○ g)(x) = f(g(x)) - The domain of f ○ g is all x-values in the domain of g that map to g(x)-values in the domain of f.

  6. f○g G(x) f g x F(g(x)) And g(x) must be in the domain of f X must be in the domain of g

  7. Composing Functions: • Let f(x) = ex and g(x) = √x • Find (f ○ g)(x) and (g ○ f)(x) and verify that the functions f ○ g and g ○ f are not the same.

  8. Finding the domain of a Composition: • Let f(x) = x2 - 1 and let g(x) = √x. Find the domains of the composite functions: (b) f ○ g (a) g ○ f

  9. Decomposing functions: For each function h, find functions f and g such that h(x) = f(g(x)) h(x) = (x+4)2 - 3(x+4) + 4 h(x) = √(x3 +1)

  10. Inverse Relations and Inverse Functions: Inverse Relation: The ordered pair (a,b) is in a relation IFF the ordered pair (b,a) is in the inverse relation. Inverse Function: If f is a one-to-one function with domain D and range R, then the inverse function of f, f-1, is the function with domain D and range R defined by F-1 (b) = a IFF f(a)=b

  11. Finding an Inverse function algebraically: F(x) = x/(x+1) Find f-1(x)

  12. Do Now: Find the inverse: F(x) = 3√(x -6) F(x) = (3x - 5)/(x+4) F(x) = x2 + 7 Homework: Pg. 129 #44-52

  13. …. Find f-1(x) of the following functions: F(x) = 3x - 6 F(x) = √(x+5) F(x) = 3√(x -1) F(x) = (2x - 4)/(x+2) F(x) = x2

  14. The Inverse Reflection Principle: The points (a,b) and (b,a) in the coordinate plane are symmetric to the line y=x. The points (a,b) and (b,a) are reflections of each other across the line y=x.

  15. The Inverse Composition Rule: A function f is one-to-one with inverse function g IFF: F(g(x)) = x for every x in the domain of g, and g(f(x)) = x for every x in the domain of f Verify inverse function: F(x) = x3 + 1 and g(x) = 3√(x-1)

  16. Find the inverse of the following function: F(x) = √(x+3) Find the domain and range of f and f-1

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