Pre-Algebra Examples

$g (x) = - \sqrt{3 x + 4}$

Step 1

Find the domain for $y = - \sqrt{3 x + 4}$ 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 $\sqrt{3 x + 4}$ greater than or equal to $0$ to find where the expression is defined.

$3 x + 4 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $4$ from both sides of the inequality.

$3 x \geq - 4$

Step 1.2.2

Divide each term in $3 x \geq - 4$ by $3$ and simplify.

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Step 1.2.2.1

Divide each term in $3 x \geq - 4$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 4}{3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 4}{3}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 4}{3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{4}{3}$

Step 1.3

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

Interval Notation:

$[- \frac{4}{3}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{4}{3}}$

Interval Notation:

$[- \frac{4}{3}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{4}{3}}$

Step 2

To find the radical expression end point, substitute the $x$ value $- \frac{4}{3}$ , which is the least value in the domain, into $f (x) = - \sqrt{3 x + 4}$ .

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Step 2.1

Replace the variable $x$ with $- \frac{4}{3}$ in the expression.

$f (- \frac{4}{3}) = - \sqrt{3 (- \frac{4}{3}) + 4}$

Step 2.2

Simplify the result.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$f (- \frac{4}{3}) = - \sqrt{3 (\frac{- 4}{3}) + 4}$

Step 2.2.1.2

Cancel the common factor.

$f (- \frac{4}{3}) = - \sqrt{3 (\frac{- 4}{3}) + 4}$

Step 2.2.1.3

Rewrite the expression.

$f (- \frac{4}{3}) = - \sqrt{- 4 + 4}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Add $- 4$ and $4$ .

$f (- \frac{4}{3}) = - \sqrt{0}$

Step 2.2.2.2

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

$f (- \frac{4}{3}) = - \sqrt{0^{2}}$

Step 2.2.2.3

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

$f (- \frac{4}{3}) = - 0$

Step 2.2.2.4

Multiply $- 1$ by $0$ .

$f (- \frac{4}{3}) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(- \frac{4}{3}, 0)$ .

$(- \frac{4}{3}, 0)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 4.1

Substitute the $x$ value $- 1$ into $f (x) = - \sqrt{3 x + 4}$ . In this case, the point is $(- 1, - 1)$ .

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Step 4.1.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - \sqrt{3 (- 1) + 4}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $3$ by $- 1$ .

$f (- 1) = - \sqrt{- 3 + 4}$

Step 4.1.2.2

Add $- 3$ and $4$ .

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

Step 4.1.2.3

Any root of $1$ is $1$ .

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

Step 4.1.2.4

Multiply $- 1$ by $1$ .

$f (- 1) = - 1$

Step 4.1.2.5

The final answer is $- 1$ .

$y = - 1$

Step 4.2

Substitute the $x$ value $0$ into $f (x) = - \sqrt{3 x + 4}$ . In this case, the point is $(0, - 2)$ .

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Step 4.2.1

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

$f (0) = - \sqrt{3 (0) + 4}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $3$ by $0$ .

$f (0) = - \sqrt{0 + 4}$

Step 4.2.2.2

Add $0$ and $4$ .

$f (0) = - \sqrt{4}$

Step 4.2.2.3

Rewrite $4$ as $2^{2}$ .

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

Step 4.2.2.4

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

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

Step 4.2.2.5

Multiply $- 1$ by $2$ .

$f (0) = - 2$

Step 4.2.2.6

The final answer is $- 2$ .

$y = - 2$

Step 4.3

Substitute the $x$ value $1$ into $f (x) = - \sqrt{3 x + 4}$ . In this case, the point is $(1, - \sqrt{7})$ .

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Step 4.3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \sqrt{3 (1) + 4}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Multiply $3$ by $1$ .

$f (1) = - \sqrt{3 + 4}$

Step 4.3.2.2

Add $3$ and $4$ .

$f (1) = - \sqrt{7}$

Step 4.3.2.3

The final answer is $- \sqrt{7}$ .

$y = - \sqrt{7}$

Step 4.4

The square root can be graphed using the points around the vertex $(- 1.\overline{3}, 0), (- 1, - 1), (0, - 2), (1, - 2.65)$

$\begin{array}{c} x & y \\ - 1.333 & 0 \\ - 1 & - 1 \\ 0 & - 2 \\ 1 & - 2.65 \end{array}$

Step 5