Operations on Functions

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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

xgxfxgf This just says that to find the sum of two functions, add them together. You should simplify by finding like terms.

1432 32 xxgxxf

1432 32 xxgf

424 23 xx

Combine like terms & put in descending

The difference f - g

xgxfxgf 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.

1432 32 xxgxxf

1432 32 xxgf

1432 32 xx

Distribute negative

224 23 xx

The product f • g

xgxfxgf 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.

1432 32 xxgxxf

1432 32 xxgf

32128 235 xxx

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.

1432 32 xxgxxf

f 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.

COMPOSITION

OFFUNCTIONS

“SUBSTITUTING ONE FUNCTION INTO ANOTHER”

The Composition Function

xgfxgf 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.

1432 32 xxgxxf

314223 xgf

51632321632 3636 xxxx

FOIL first and then distribute

xfgxfg 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.

1432 32 xxgxxf

132432 xfg

You could multiply this out but since it’s to the 3rd power we

won’t

xffxff 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).

1432 32 xxgxxf

332222 xff

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.

1 is ofdomain xxgf

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.)

Operations on Functions

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