Performing operations on functions allows us to review algebraic manipulations such as adding and subtracting like terms, simplifying algebraic expressions, the distributive property of multiplication, factoring and long division. Be sure you understand all these operations before watching this Crash Course.
The operations on functions include addition (sum), subtraction (difference), multiplication (product), division (quotient) and composition.
When we add functions, only the like terms are added.
When we subtract functions, we distribute the subtraction correctly to all the terms of the 2nd function, then group the like terms together.
When we multiply functions, we distribute every term of the 1st function to every term of the 2nd function, then group the like terms together.
When dividing functions, you can choose to use either of the following 2 strategies:
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Completely factor the dividend (numerator) and the divisor (denominator), then simplify the common factors.
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Perform long division.
When performing the composition of functions |g \circ f,| we substitute the independent variable of function |g| by the algebraic expression that represents function |g.| ||(g\circ f)(x) = g\big(f(x)\big)||
Notes:
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The domain of the resulting function corresponds to the intersection of the starting function domains.
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When dividing functions, we must also exclude from the domain any |x| value that makes the divisor (denominator) equal to |0.|
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Subtraction, division and the composition of functions are not commutative.