Checking Validity of Functions and Inverse Functions

Sometimes it can be hard to tell whether or not a graph or equation is a valid function or not. Here are some simple steps to checking this:

Horizontal line test: If a straight line can be drawn horizontally that crosses the graph in two places, then this function is called a many to one function. This is only true if a values of x are plotted. However, when finding the inverse of this function, the graph will be reflected in y=x and will no longer abide by the Horizontal line test, but the Vertical line test.

Vertical Line test: If a straight, vertical line can be drawn that crosses the graph in two places, then this is 'one to many' which is not a function. 

1) Vertical line test true = not a function.

2) Horizontal line test true = many to one function

    Horizontal line test true = inverse will not be a function.

Long Division VS Remainder Theorem

Both methods may be used to divide polynomials by a factor such as (x-3), however take different steps in reaching the end goal.


Long Division

As shown, the factor (x-3) is multiplied by a term to create a new expression that can be subtracted from the original polynomial. This is repeated until the polynomial reaches 0 in which case there is no remainder, or it cannot be divided further, in which case there is a remainder.
The terms are then collected and the remainder is stated as above: 29/(x-3) where the remainder is divided by the divisor.

The Remainder Theorem

This method in my opinion takes longer. The idea is that a polynomial (Ax^3 + Bx^2 + Cx + D) is created and using substitution, these values of ABCD can be calculated. First assume there is a remainder i.e. +D.
We then let x =3. This allows us to work out the value of D.
Then we let x =0. This allows us to work out the value of C.
We then compare coefficients i.e. the coefficient (LHS) of x^3 is 1. Therefore the coefficients of all the x^3's on the RHS must also equal to 1. Therefore A=1.
With x^2 LHS equals 1. Therefore all of the x^2 terms i.e. -3Ax^2 + Bx^2 must also equal 1. This can be used to work out the remaining values of ABCD and the remainder stated, divided by the divisor i.e. 29/(x-3).