Understanding Composite Functions in Mathematics
Learn about composite functions in mathematics, where one function is followed by another to create a new function. Explore examples and how to find composite function outputs. Discover how the order of functions impacts the final result and understand the domains of composite functions.
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Presentation Transcript
Composite functions When a function is followed by another, the result is a composite function output input Function Machine f Function Machine g
Composite functions If g: x 2x and f: x x + 3, then 7 2 Function Machine f x + 3 Function Machine g 2x 4 4
Composite functions If x is the input, then f [g(x)] x Function Machine f Function Machine g g(x) g(x) f [g(x)] is usually written without the square brackets as fg(x) fg(x) means do g first followed by f Note that the domain of f is the range of g
If the order of the functions change If g: x 2x and f: x x + 3, then 10 2 Function Machine g 2x Function Machine f x + 3 5 5
If the order of the functions change g[f(x)] x Function Machine g Function Machine f f(x) f(x) g[f(x)] is usually written without the square brackets as gf(x) gf (x) means do f first followed by g Note that the domain of g is the range of f
Composite functions If f (x) = x2 and g(x) = x + 3 a) find: fg (2) g (2) = 2 + 3 g (2) = 5 f (5) = 52 f (5)= 25 = 25 fg (2) do g first followed by f b) gf (2) f (2) = 22 f (2) = 4 g (4) = 4 + 3 g (4) = 7 gf (2) = 7 do f first followed by g
Composite functions If f (x) = x2 and g(x) = x + 3 c) find: fg (x) g (x) = x + 3 do g first f (x + 3) = (x + 3)2 followed by f fg (x) = (x + 3)2 gf (x) f (x) = x2 d) do f first g (x2)= x2 + 3 followed by g gf (x) = x2 + 3
which is the domain of: If f (x) = x - 2 and g(x) = 3x a) gf (x)? = x - 2 f (x) g (x - 2) = 3(x - 2) 3(x - 2) 0 3x - 6 0 3x 6 x 2 which means the domain is {x:x 2, x a real number}
which is the domain of: If f (x) = x - 2 and g(x) = 3x b) fg (x)? = 3x g (x) f ( ) 3x = 3x - 2 3x 0 x 0 which means the domain is {x:x 0, x a real number}
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