Examples

$f (x) = x - 6$ , $(0, 7)$

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:

${x | x \in ℝ}$

Step 3

Subtract $6$ from $0$ .

$f (0) = - 6$

Step 4

Subtract $6$ from $7$ .

$f (7) = 1$

Step 5

Since $0$ is on the interval $[- 6, 1]$ , solve the equation for $x$ at the root by setting $y$ to $0$ in $y = x - 6$ .

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Step 5.1

Rewrite the equation as $x - 6 = 0$ .

$x - 6 = 0$

Step 5.2

Add $6$ to both sides of the equation.

$x = 6$

Step 6

The Intermediate Value Theorem states that there is a root $f (c) = 0$ on the interval $[- 6, 1]$ because $f$ is a continuous function on $[0, 7]$ .

The roots on the interval $[0, 7]$ are located at $x = 6$ .

Step 7

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