Fundamental Theory of Algebra

The fundamental theory of algebra states that any polynomial of degree n has n roots. For example, the function  has seven roots, because the highest exponent is seven. Those roots are . Five of these roots are real, but two are unreal. This can be seen in the graph:

Again, this function has five, real, visible roots and two unreal ones that cannot be easily seen on the graph. The roots we cannot see as x-intercepts can be assumed to be imaginary. This can be demonstrated in the translation of the function .

Four real roots, zero imaginary roots. 

Two reals roots, two imaginary roots.

Zero real roots, four imaginary roots.

As you probably have noticed, the roots in this function come in pairs. Real roots only come in pairs in a function that is symmetrical about the y-axis, but imaginary roots always come in pairs, no matter the shape. This is because imaginary roots come in conjugate pairs. For example, if F(2+3i)=0 then, F(2-3i)=0. 

Arithmetic Combinations of Functions

Arithmetic Combinations of Functions
We know that functions are a relationship between two unknowns like a ratio is a relationship between two numbers. Relationships between two numbers can be arithmetically combined, ex.

So functions can be combined in a similar way.

What does that mean?:

Arithmetic combinations of functions are the sum, difference, product, or quotient of two functions,  commonly f(x) and g(x), with overlapping domains. That's right they must have overlapping domains (where both functions are real and defined). The operation is just simple arithmetic but be on the lookout for the distributive property and accidentally simplifying parts of expressions.

For each example of the types of arithmetic combinations of functions, we will use: 

Sum of Functions:

Defined as:

The sum of the function f and the function g is equal to the sum of their outputs, respectively.

As for our example, simply add g(x) to f(x):

We could also evaluate an input in this new compound function, which is named (f + g)(x), by inputting a value for x. Let's evaluate (f + g)(1):

If we look at a graphical approach to the sum of functions, it gets a little more complicated. For every value of f(x), we have to add the corresponding value of g(x). 

f(x):

                                                           (f+g)(x):

g(x):

We can see that the graph looks to be exactly what the name (f+g)(x) describes: every y value of g(x) was added to the corresponding value of f(x).

Difference of Functions:

Defined as:

The difference of f(x) and g(x) is equal to the difference of the outputs, respectively.

Example (just subtract g(x) from f(x)):

Now let's evaluate (f - g)(0):

Graphically, (f - g)(x) is the y values of the graph  f(x) minus the y values of g(x), as we can see here:

 f(x):

                                                           (f - g)(x):







g(x):

We can see that all of the graph of f(x) was lowered except where the y value of g(x) is equal to zero.

Product of Functions:

Defined as:

The product of the the functions f(x) and g(x) is equal to the product of their outputs.

In our example, the produce of f(x) and g(x) is found be the following process:

Evaluate (fg)(-1):

In graphing (fg)(x), we would go about it in a similar fashion as before: multiply the output of f(x) by the output of g(x).

f(x):

                                                            (fg)(x)

g(x):

In these graphs, the results are a little extreme but you can still tell that when the outputs of both f(x) and
g(x) are the same sign, the final output is positive. When f(x) and g(x) are opposite signs, the result is negative.

Quotient of Functions:

Defined by:

The quotient of f(x) and g(x) is equal to the output of f(x) divided by the output of g(x).

Following the example above:






And evaluating (f / g)(2):

The graph of (f / g)(x) is has a catch that all others do not. If g(x) is not constant, then the zeroes of g(x) must be excluded from the domain of the function. This is because if g(x) is zero, the whole rational function is undefined. Many problems will ask for a domain along with the equation for (f / g)(x). For our example:

So we expect a vertical asymptote when x equals negative one half. Which is reflected by the graph of
(f / g)(x).

f(x):





                                                           (f / g)(x):



g(x):

General Tips:

1. (f + g), (f - g), (fg), and (f / g) are names not operations, don't let them confuse you!
2. If you are doing subtraction, always put the g(x) in parenthesis and distribute the negative.
3. For (f / g)(x), exclude the values where g(x)=0 because that makes the function undefined.
4. Never simplify expressions. If you have something like the following example, you must factor to simplify.