class opener:. identifying a composite function:
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Class Opener:
• Given: and find each composition:a) fog(x)b) gof(x)
What is the domain of (fog)(x) given:
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Identifying a Composite Function:
• Write the following function as a composition of two functions:
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Identifying a Composite Function:
• Write the following function as a composition of two functions:
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Identifying a Composition of Two Functions:
• Find two functions f and g such that
Where:
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ORQ Practice: Bacteria Count:
The number N of bacteria in a refrigerated food is given by
Where T is the temperature of the food. When the food is removed from the refrigeration, the temperature of the food is given by
T(t) = 4t + 2, Where t is the time in hours.
a) Find the composition N(T(t)) and interpret its meaning in context.
b) Find the number of bacteria in the food when t = 2c) Find the time when the bacterial count reaches 2000
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Inverse Functions:
• An inverse function is a function from Set B to Set A, and is denoted by
• The domain of the original function f is equal to the range of , and vice versa.
• The composition of f and will result in the identify function.
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Finding the Inverse Function Informally
• Find the inverse function of:
• Prove that the inverse function and the original function will produce the identify function.
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Finding the Inverse Function Informally
• Find the inverse function for each:
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Verifying the Inverse Function Algebraically
• Show that the functions are invers functions of each other:
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Verifying the Inverse Function Algebraically
• Which of the functions is the inverse function of
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One – to – One Functions
• A function is one to one if, for a and b in its domain, f(a) = f(b) implies that a = b.
• A function f has an inverse function if and only if f is one to one.
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Testing for one to one functions
Is the function: one to one
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Testing one to one functions
• Is the function one to one
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Horizontal Line Test
• Use the horizontal line test on a graph of a function to see if it is one to one.
• If it is a function the horizontal line will only hit the function one time.