# Finite Math Examples

Prove that a Root is on the Interval
,
The Intermediate Value Theorem states that, if is a real-valued continuous function on the interval , and is a number between and , then there is a contained in the interval such that .
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Calculate .
Simplify each term.
Remove parentheses around .
Raising to any positive power yields .
Multiply by to get .
Simplify by adding zeros.
Add and to get .
Subtract from to get .
Calculate .
Simplify each term.
Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Simplify by adding and subtracting.
Add and to get .
Subtract from to get .

Rewrite the equation as .
Graph each side of the equation. The solution is the x-value of the point of intersection.
The Intermediate Value Theorem states that there is a root on the interval because is a continuous function on .
The roots on the interval are located at .