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.
Interval Notation:
Set-Builder Notation:
Raise to the power of .
Raise to the power of .
Since is on the interval , solve the equation for at the root by setting to in .
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Rewrite the equation as .
Take the cube root of both sides of the equation to eliminate the exponent on the left side.
Simplify .
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Rewrite as .
Pull terms out from under the radical, assuming real numbers.
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 .
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