Finite Math Examples

$y = 3^{x}$ , $[- 3, 3]$

Step 1

The Intermediate Value Theorem states that, if $f$ is a real-valued continuous function on the interval $[a, b]$ , and $u$ is a number between $f (a)$ and $f (b)$ , then there is a $c$ contained in the interval $[a, b]$ such that $f (c) = u$ .

$u = f (c) = 0$

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:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 3

Calculate $f (a) = f (- 3) = 3^{- 3}$ .

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f (- 3) = \frac{1}{3^{3}}$

Step 3.2

Raise $3$ to the power of $3$ .

$f (- 3) = \frac{1}{27}$

Step 4

Raise $3$ to the power of $3$ .

$f (3) = 27$

Step 5

$0$ is not on the interval $[\frac{1}{27}, 27]$ .

There is no root on the interval.

Step 6