Algebra of Functions - Pre-Calculus

First, make sure that gf (range of g is a subset of the domain of f).

Since the g: and f: , gf and exists.

Plug in the output of , which is , as the input of .

Thus,

First, make sure that gf (range of g is a subset of the domain of f).

Since the g: and f: , gf and exists.

Plug in the output of , which is , as the input of .

Thus,

First, make sure that gf (range of g is a subset of the domain of f).

Since the g: and f: , gf and exists.

Plug in the output of , which is , as the input of .

Thus,

To add, simply combine like terms. Thus, the answer is:

When adding two expressions, you can only combine terms that have the same variable in them.

In this question, we get:

Now we can add each of the results to get the final answer:

The first step is to rewrite the expression:

Now that it is expanded, we can FOIL (First, Outer, Inner, Last) the expression:

First :

Outer:

Inner:

Last:

Now we can simply add up the values to get the expanded expression:

First, we can start off by factoring out constants from the numerator and denominator.

The 9/3 simplifies to just a 3 in the numerator. Next, we factor the top numerator into , and simplify with the denominator.

We now have

First, distribute the -5 to each term in the second expression:

Next, combine all like terms

to end up with

.

We begin by factoring and we get .

Now, When we look at it will be .

We can take out from the numerator and cancel out the denominator, leaving us with .

First, we must determine what is equal to. We do this by distributing the 3 to every term inside the parentheses,.

Next we simply subtract this from , going one term at a time:

Finally, combining our terms gives us .

In order to fully expand the expression , let's first rewrite it as:

.

Then, using the FOIL(First, Outer, Inner, Last) Method of Multiplication, we expand the expression to:

First:

Outer:

Inner:

Last:

, which in turn

To simplify the above expression, we must combine all like terms:

:

:

:

Integers:

Putting all of the above terms together, we simplify to:

Given the information in the above problem, we know that:

Factoring the resulting fraction, we get:

To simplify the expression, distribute the negative into the second parentheses, and then combine like terms.

To simlify adding polynomials, simply drop the parentheses and add like terms.

To add the numerators, the denominators must be common.

The least common denominator can be determined by multiplication.

Rewrite the fractions.

Given and

Complete the operation given by .

Begin by realizing what this is asking. We need to combine our two functions in such a way that we find the difference between them.

When doing so remember to distribute the negative sign that is in front of to each term within the polynomial.

So, by simplifying the expression, we get our answer to be:

To solve , simply multiply your two functions. Thus,

Given and ,

Evaluate and simplify .

Begin by multiplying by 2:

Next, add to what we got above and combine like terms.

This makes our answer

.