1.6 - Combinations of Functions
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The domain of each of these combinations is the intersection of the domain of f and the domain of g. In other words, both functions must be defined at a point for the combination to be defined. One additional requirement for the division of functions is that the combining functions and their domains can't be zero, but we knew that because it's part of the implied domain. Basically what the above says is that to evaluate a combination of functions, you may combine the functions and then evaluate or you may evaluate each function and then combine.
As you can see from the examples, it doesn't matter if you combine and then evaluate or if you evaluate and then combine. In each of the above problems, the combining functions and their domains is all real numbers with the exception of the division. The domain in the division combination is all real numbers except for 1 combining functions and their domains While the arithmetic combinations of functions are straightforward and fairly easy, there is another type of combination called a composition.
A composition of functions is the applying of one function to another function. The symbol of composition of functions is a small circle between the function names. I can't do that combining functions and their domains in text mode on the web, so I'll use a lower case oh " o " to represent composition of functions.
These combining functions and their domains read "f composed with g of x" and "g composed with f of x" respectively. The function on the outside is always written first with the functions that follow being on the inside. The order is important. Composition of functions is not commutative. This example probably needs some explanation. From the prerequisite chapterthe square root of x 2 is the absolute value of x. The square of the square root of x is x, but this assumes that x is not negative because you couldn't find the square root of x in the first place if it was.
This is a case where the implied domain because of the square root is no longer implied because the square root is goneso you have to explicitly state it I told you it all fit together.
If the last example needed some explanation, then this one definitely needs some, too. Let's take the easier one g o combining functions and their domains x first. Okay, now for the harder one f o g x. I'll give the simple explanation here and the more complete one later.
After simplifying, you got the square root of -x 2 - 3. When you find a composition of a functions, it is no longer x that combining functions and their domains being plugged into the outer function, it is the inner function evaluated at x. So there are two domains that we have to be concerned about. If we consider f o g xwe see that g is evaluated at x, so x has to be in the domain of g.
We also see that f is evaluated at g xso g x has to be in combining functions and their domains domain of f. When you combine the two domains to see what they have in common, you find the intersection of everything combining functions and their domains nothing is nothing the empty setso the function is defined nowhere and undefined everywhere. Decomposition of functions is the reverse of composition of functions. Instead of combining two functions to get a new function, you're breaking apart a combined function into its separate components.
There is often more than one way to decompose a function, so your answers may vary from the books. Basically, you want to look at the function and look for an "outside function" and an "inside function". Another thing to look for is repeated patterns and make that the inside function. The outside function is summarized as "the big picture" and the inside function is "what you are doing the big picture to".
Difference quotients are what they say they are. They involve a difference and a quotient. A difference quotient is really the slope of a secant line between two points on a curve. For polynomial functions, finding the difference quotient isn't that difficult. Where you're going to run into trouble is combining functions and their domains radical and rational functions. The trick with a radical function is to rationalize the numerator by multiplying by the conjugate of the numerator.
Don't worry that you're left with a radical in the denominator, it's okay in this instance. It's much better than having a factor of h in the denominator because in calculus, we're going to let h approach 0 and we'll want to just plug a zero in for h. When given the choice of having a radical in the denominator or division by 0, we'll pick a radical in the denominator anyday. The big thing going on is cubing something, so the outside function is a cubing function.
The big thing going on is taking the square root outside9-x is what you're taking the square root of inside. Combining functions and their domains every occurrence of the pattern by x for the outside function.