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Composite Functions Form a Composite Functions and its Domain

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Composite Function Given two functions f and g, the composite function, denoted by f ◦ g (read as f composed with g) is defined by (f ◦ g)(x) = f(g(x)) The domain of f ◦ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

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Evaluating a Composite Function Example 1 Suppose that f(x) = 2x 3 -3 and g(x) = 4x Find a.(f ◦ g)(1)= b.(g ◦ f)(1) = c.(f ◦ f)(-2) = d.(g ◦ g) (-1) =

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Finding a Composite Function Example 2 Suppose that f(x)= x 2 +3x-1 and g(x) = 2x+3 Find a.f ◦ g and state the domain of each composite function

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Finding a Composite Function Example 3 f(x)= x 2 +3x-1 and g(x) = 2x+3 g ◦ f and state the domain of each composite function

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Finding the Domain of f ◦ g Find the domain (f ◦g )(x) if f(x) = and g(x) =

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Solution Domain of g is {x │x ≠ 1} Therefore 1 has to be excluded from the domain of the composite Domain of f is {x│x≠ -2} This means g(x) can never equal -2 we need to solve for g(x) = -2

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Finding a Composite Function Suppose that f(x)= and g(x) = Find f ◦ g Find f ◦ f

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Showing that two Composite Functions are equal If f(x) = 3x-4 and g(x) = show that (f ◦ g)(x) = (g ◦ f)(x)= x

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Homework Page 229 examples 7-45 every third question

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Inverse Functions Determine the Inverse of a Function Obtain the Graph of the Inverse Function from the graph of the Function Find the Inverse Function f -1

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What is the Inverse? Something we know: We know that a function can be compared to a machine. We input an x and the machine manipulates it and outputs the value f(x) or the y value. The inverse of f receives as input a number f(x) and manipulates it and outputs the value x.

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Finding the inverse of a Function Let the domain of the function represent the people we know and let the range represent their birthdays. DomainRange Mrs IrelandDecember 7 Bry’EishaMarch 11 MelissaOctober 7 AllenaMarch 1

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Finding the inverse of a function Find the inverse of the following functions: {(-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4), (3,9)} The inverse of the given function is found by interchanging the entries in each ordered pair Answer: {(9,-3),(4,-2),(1,-1),(0,0),(1,1),(4,2)(9,3) Is the inverse a function? No because there are two outputs for 4, 9 and 1

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Finding the inverse of a function Example 2: Find the inverse of the function {(-3,-27),(-2,-8),(-1,-1),(0,0),(1,1),(2,8),(3,27) Answer: {(-27,-3),(-8,-2),(-1,-1),(0,0),(1,1),(8,2),(27,3) Is the inverse a function? Yes! Why?

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One to one When the inverse of a function f is itself a function then f is said to be a one to one function. That is f is one to one if for any choice of elements x 1 and x 2 in the domain of f, with x 1 ≠ x 2, the corresponding values f(x 1 ) and f(x 2 ) are unequal, f(x 1 ) ≠ f(x 2 ) In words A function is one to one if any two different inputs never correspond to the same output

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Horizontal Line Test Something we know: We know to test whether a graph is a function we use the vertical line test Now we have a new test Horizontal Line Test: If every horizontal line intersects the graph of a function f in at most one point, then f is one to one.

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