Finite Math Examples

f (x) = \frac{1}{{(x - 0.78)}^{2} + 0.024}

$f (x) = \frac{1}{{(x - 0.78)}^{2} + 0.024}$

Step 1

Set the denominator in

\frac{1}{{(x - 0.78)}^{2} + 0.024}

$\frac{1}{{(x - 0.78)}^{2} + 0.024}$ equal to

0

$0$ to find where the expression is undefined.

{(x - 0.78)}^{2} + 0.024 = 0

${(x - 0.78)}^{2} + 0.024 = 0$

Step 2

Solve for

x

$x$ .

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Step 2.1

Subtract

0.024

$0.024$ from both sides of the equation.

{(x - 0.78)}^{2} = - 0.024

${(x - 0.78)}^{2} = - 0.024$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

x - 0.78 = \pm \sqrt{- 0.024}

$x - 0.78 = \pm \sqrt{- 0.024}$

Step 2.3

Simplify

\pm \sqrt{- 0.024}

$\pm \sqrt{- 0.024}$ .

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Step 2.3.1

Rewrite

- 0.024

$- 0.024$ as

- 1 (0.024)

$- 1 (0.024)$ .

x - 0.78 = \pm \sqrt{- 1 (0.024)}

$x - 0.78 = \pm \sqrt{- 1 (0.024)}$

Step 2.3.2

Rewrite

\sqrt{- 1 (0.024)}

$\sqrt{- 1 (0.024)}$ as

\sqrt{- 1} \cdot \sqrt{0.024}

$\sqrt{- 1} \cdot \sqrt{0.024}$ .

x - 0.78 = \pm \sqrt{- 1} \cdot \sqrt{0.024}

$x - 0.78 = \pm \sqrt{- 1} \cdot \sqrt{0.024}$

Step 2.3.3

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

x - 0.78 = \pm i \sqrt{0.024}

$x - 0.78 = \pm i \sqrt{0.024}$

x - 0.78 = \pm i \sqrt{0.024}

$x - 0.78 = \pm i \sqrt{0.024}$

Step 2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.4.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

x - 0.78 = i \sqrt{0.024}

$x - 0.78 = i \sqrt{0.024}$

Step 2.4.2

Add

0.78

$0.78$ to both sides of the equation.

x = i \sqrt{0.024} + 0.78

$x = i \sqrt{0.024} + 0.78$

Step 2.4.3

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

x - 0.78 = - i \sqrt{0.024}

$x - 0.78 = - i \sqrt{0.024}$

Step 2.4.4

Add

0.78

$0.78$ to both sides of the equation.

x = - i \sqrt{0.024} + 0.78

$x = - i \sqrt{0.024} + 0.78$

Step 2.4.5

The complete solution is the result of both the positive and negative portions of the solution.

x = i \sqrt{0.024} + 0.78, - i \sqrt{0.024} + 0.78

$x = i \sqrt{0.024} + 0.78, - i \sqrt{0.024} + 0.78$

x = i \sqrt{0.024} + 0.78, - i \sqrt{0.024} + 0.78

$x = i \sqrt{0.024} + 0.78, - i \sqrt{0.024} + 0.78$

x = i \sqrt{0.024} + 0.78, - i \sqrt{0.024} + 0.78

$x = i \sqrt{0.024} + 0.78, - i \sqrt{0.024} + 0.78$

Step 3

The domain is all real numbers.

Interval Notation:

(- \infty, \infty)

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4