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Understanding Functions, Composition, and Inverses in Mathematics

This guide explores the fundamentals of functions, including their definitions, operations (addition, subtraction, multiplication, and division), and composition denoted as fog and gof (f◦g and g◦f). It presents examples to illustrate these concepts, such as evaluating function operations and determining the domain of composite functions. Additionally, it explains how to find the domain of compositions to ensure valid outputs. This foundational knowledge is crucial for further studies in algebra and calculus.

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Understanding Functions, Composition, and Inverses in Mathematics

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  1. 4-4 Functions Fogs, gofs and inverses

  2. What is a function? A set of (x,y) pairs where each x has a unique y value. Functions can be added, subtracted, multiplied and divided.

  3. Examples 1. If and find c) f(x) ÷ g(x) a) f(x) + g(x) b) f(x) · g(x) 2. For the above functions find a) f(3) · g(3) b) Plug 3 into 1b (above) Note: When you add, subtract, multiply and divide functions, the domain of the result is the domain common to the functions that you used.

  4. Fog? Gof? What is that? “fog” is the nickname for f◦g and “gof” is the nickname for g◦f. That notation indicates “composition”. That is, we are taking the composition of 2 (or more) functions. You have seen composition before but it isn’t something that jumps out at you.

  5. This is a composition can be thought of as the composition of the functions and You sort of “tuck” one function into the other.

  6. If and Then Now, what would g◦f be?

  7. Examples 4. If and find a) f(x) ◦ g(x) b) g(x) ◦ f(x) • If , and • find f ◦ g ◦ h. 6. If and find f ◦ g

  8. Domain of a composite The domain of a composite is all x in the domain of the inner function for which the values are in the domain of the outer function. Just determine domain of inner function, then exclude anything that is not in the domain of the outer function. Note: Please add #59 to the homework assignment

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