Pre-Algebra Examples

$f (x) = 2 \sqrt{- 2 x} - 1$

Step 1

Find the domain for $y = 2 \sqrt{- 2 x} - 1$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

Set the radicand in $\sqrt{- 2 x}$ greater than or equal to $0$ to find where the expression is defined.

$- 2 x \geq 0$

Step 1.2

Divide each term in $- 2 x \geq 0$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 x \geq 0$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{0}{- 2}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{0}{- 2}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{0}{- 2}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $- 2$ .

$x \leq 0$

Step 1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${x | x \leq 0}$

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${x | x \leq 0}$

Step 2

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = 2 \sqrt{- 2 x} - 1$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 2 \sqrt{- 2 \cdot 0} - 1$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $- 2$ by $0$ .

$f (0) = 2 \sqrt{0} - 1$

Step 2.2.1.2

Rewrite $0$ as $0^{2}$ .

$f (0) = 2 \sqrt{0^{2}} - 1$

Step 2.2.1.3

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = 2 \cdot 0 - 1$

Step 2.2.1.4

Multiply $2$ by $0$ .

$f (0) = 0 - 1$

Step 2.2.2

Subtract $1$ from $0$ .

$f (0) = - 1$

Step 2.2.3

The final answer is $- 1$ .

$- 1$

Step 3

The radical expression end point is $(0, - 1)$ .

$(0, - 1)$

Step 4

The square root can be graphed using the points around the vertex $(0, - 1),$

$\begin{array}{c} x & y \\ 0 & - 1 \end{array}$

Step 5