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Function Operations

Function Operations. Sum: (f+g)(x) = f(x) + g(x) Difference : (f – g) (x) = f(x) – g(x) Product: (fg)(x) = f(x)g(x) Quotient: . These operations build a new function – with a domain and range. - Will the domain and range always be the same as that of f(x) and/or g(x)?.

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Function Operations

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  1. Function Operations • Sum: (f+g)(x) = f(x) + g(x) • Difference: (f – g)(x) = f(x) – g(x) • Product: (fg)(x) = f(x)g(x) • Quotient: • These operations build a new function – with a domain and range. • - Will the domain and range always be the same as that of f(x) and/or g(x)?

  2. Composition of Functions • Composition of two functions, f and g, written as: • The range of g intersects the domain of f. • The domain of f(g(x)) consists of all the values in the domain of g that map to g(x) values in the domain of f. • In other words, in order for a number to be in the domain of f(g(x)) , it must first be in the domain of g(x). • AND, f(g(x)) only gets to “work on” numbers that are in the range of g(x) .

  3. RANGE of g acts as the DOMAIN of f(g(x)) g(x) DOMAIN of g x f(g(x))

  4. Example: Function Composition What would g(f(x)) look like? And then, “g- it” Start with what is being “acted on” Now simplify:

  5. Composition of two functions The domain is all real numbers except -1. The domain is nonzero real numbers. The domain is nonzero real numbers.

  6. Find each of the following and state the domain: The domain of f(g(x)) is all real numbers except 1 or -1 The domain of g(f(x)) is all real numbers except 0 The domain of f(f(x)) is all real numbers except 0

  7. You try!!!!!

  8. Implicitly DefinedRelations • The general term for a set of ordered pairs, (x, y) is a RELATION • Examine this RELATION • y and x are related to one another Do you know what this graph looks like? This relation is not a function. However, we can split it into two equations that do define functions. Solving for y yields two equations:

  9. Check it out… Why doesn’t it look like a circle???

  10. The window is rectangular!But you can fix it…

  11. Example of Implicitly and Explicitly Defined Functions IMPLICITLY DEFINED EXPLICITLY DEFINED

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