Power Functions A power function has the form a is a real number b is a rational number ( if b is a positive integer then a power function is a type of polynomial function.)

The sum f + g This just says that to find the sum of two functions, add them together. You should simplify by finding like terms. Combine like terms & put in descending order

The difference f - g To find the difference between two functions, subtract the first from the second. CAUTION: Make sure you distribute the – to each term of the second function. You should simplify by combining like terms. Distribute negative

The product f g To find the product of two functions, put parenthesis around them and multiply each term from the first function to each term of the second function. FOIL Good idea to put in descending order.

The quotient f /g To find the quotient of two functions, put the first one over the second. Nothing more you could do here. (If you can reduce these you should).

So the first 4 operations on functions are pretty straight forward. The rules for the domain of functions would apply to these combinations of functions as well. The domain of the sum, difference or product would be the numbers x in the domains of both f and g. For the quotient, you would also need to exclude any numbers x that would make the resulting denominator 0.

All non-negative real numbers (does include zero) All positive real numbers x>0 (doesn’t include zero)

COMPOSITION OF FUNCTIONS “SUBSTITUTING ONE FUNCTION INTO ANOTHER”

The Composition Function This is read “f composition g” and means to copy the f function down but where ever you see an x, substitute in the g function. FOIL first and then distribute the 2

This is read “g composition f” and means to copy the g function down but where ever you see an x, substitute in the f function. You could multiply this out but since it’s to the 3 rd power we won’t

This is read “f composition f” and means to copy the f function down but where ever you see an x, substitute in the f function. (So sub the function into itself).

Using composition of functions A clothing store advertises that it is having a 25% off sale. For one day only, the store advertises an additional savings of 10%. A. Use a composition of functions to find the total percent discount. B. What would be the sale price of a $40 sweater?

Let x represent the price. f(x)= x -.25x =.75x g(x) = x -.10x =.90x g(f(x))=.90(.75x)=.675x.675(40)=$27

The DOMAIN of the Composition Function The domain of f composition g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. The domain of g is x  1 We also have to worry about any “illegals” in this composition function, specifically dividing by 0. This would mean that x  1 so the domain of the composition would be combining the two restrictions.

The domain of the composition function, cont. The domain of the new function, after a function operation, consists of the x values that are in the domains of both functions. Additionally, the domain of a quotient does not include x values that would make the denominator zero or that would have you take an even root of a negative number.

You must pay attention to the order of functions when they are composed. In general, f(g(x)) is not equal to g(f(x)). (The inner function is substituted into x in the outer function.)

Warm up Find : f(x) + g(x) f(x) – g(x) f(x) ◦ g(x) f(x) ÷ g(x) g(x) ÷ f(x) Solve