Finite Math Examples

Prove that a Root is on the Interval
,
Step 1
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 .
Step 2
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.
Interval Notation:
Set-Builder Notation:
Step 3
Subtract from .
Step 4
Subtract from .
Step 5
Since is on the interval , solve the equation for at the root by setting to in .
Rewrite the equation as .
Add to both sides of the equation.
Step 6
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 .
Step 7