Pre-Algebra Examples

$f (x) = \frac{\sqrt{x + 6}}{x - 4}$

Step 1

Find the $y$ value at $x = 3$ .

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Step 1.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{\sqrt{(3) + 6}}{(3) - 4}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Add $3$ and $6$ .

$f (3) = \frac{\sqrt{9}}{3 - 4}$

Step 1.2.1.2

Rewrite $9$ as $3^{2}$ .

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

Step 1.2.1.3

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

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

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Subtract $4$ from $3$ .

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

Step 1.2.2.2

Divide $3$ by $- 1$ .

$f (3) = - 3$

Step 1.2.3

The final answer is $- 3$ .

$- 3$

Step 1.3

The $y$ value at $x = 3$ is $- 3$ .

$y = - 3$

Step 2

Find the $y$ value at $x = - 5$ .

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Step 2.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \frac{\sqrt{(- 5) + 6}}{(- 5) - 4}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Add $- 5$ and $6$ .

$f (- 5) = \frac{\sqrt{1}}{- 5 - 4}$

Step 2.2.1.2

Any root of $1$ is $1$ .

$f (- 5) = \frac{1}{- 5 - 4}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Subtract $4$ from $- 5$ .

$f (- 5) = \frac{1}{- 9}$

Step 2.2.2.2

Move the negative in front of the fraction.

$f (- 5) = - \frac{1}{9}$

Step 2.2.3

The final answer is $- \frac{1}{9}$ .

$- \frac{1}{9}$

Step 2.3

The $y$ value at $x = - 5$ is $- \frac{1}{9}$ .

$y = - \frac{1}{9}$

Step 3

Find the $y$ value at $x = - 4$ .

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Step 3.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = \frac{\sqrt{(- 4) + 6}}{(- 4) - 4}$

Step 3.2

Simplify the result.

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Step 3.2.1

Add $- 4$ and $6$ .

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

Step 3.2.2

Subtract $4$ from $- 4$ .

$f (- 4) = \frac{\sqrt{2}}{- 8}$

Step 3.2.3

Move the negative in front of the fraction.

$f (- 4) = - \frac{\sqrt{2}}{8}$

Step 3.2.4

The final answer is $- \frac{\sqrt{2}}{8}$ .

$- \frac{\sqrt{2}}{8}$

Step 3.3

The $y$ value at $x = - 4$ is $- \frac{\sqrt{2}}{8}$ .

$y = - \frac{\sqrt{2}}{8}$

Step 4

Find the $y$ value at $x = 5$ .

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Step 4.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = \frac{\sqrt{(5) + 6}}{(5) - 4}$

Step 4.2

Simplify the result.

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Step 4.2.1

Add $5$ and $6$ .

$f (5) = \frac{\sqrt{11}}{5 - 4}$

Step 4.2.2

Subtract $4$ from $5$ .

$f (5) = \frac{\sqrt{11}}{1}$

Step 4.2.3

Divide $\sqrt{11}$ by $1$ .

$f (5) = \sqrt{11}$

Step 4.2.4

The final answer is $\sqrt{11}$ .

$\sqrt{11}$

Step 4.3

The $y$ value at $x = 5$ is $\sqrt{11}$ .

$y = \sqrt{11}$

Step 5

Find the $y$ value at $x = 6$ .

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Step 5.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{\sqrt{(6) + 6}}{(6) - 4}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Add $6$ and $6$ .

$f (6) = \frac{\sqrt{12}}{6 - 4}$

Step 5.2.1.2

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 5.2.1.2.1

Factor $4$ out of $12$ .

$f (6) = \frac{\sqrt{4 (3)}}{6 - 4}$

Step 5.2.1.2.2

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

$f (6) = \frac{\sqrt{2^{2} \cdot 3}}{6 - 4}$

Step 5.2.1.3

Pull terms out from under the radical.

$f (6) = \frac{2 \sqrt{3}}{6 - 4}$

Step 5.2.2

Reduce the expression by cancelling the common factors.

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Step 5.2.2.1

Subtract $4$ from $6$ .

$f (6) = \frac{2 \sqrt{3}}{2}$

Step 5.2.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.2.1

Cancel the common factor.

$f (6) = \frac{2 \sqrt{3}}{2}$

Step 5.2.2.2.2

Divide $\sqrt{3}$ by $1$ .

$f (6) = \sqrt{3}$

Step 5.2.3

The final answer is $\sqrt{3}$ .

$\sqrt{3}$

Step 5.3

The $y$ value at $x = 6$ is $\sqrt{3}$ .

$y = \sqrt{3}$

Step 6

Find the $y$ value at $x = 7$ .

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Step 6.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{\sqrt{(7) + 6}}{(7) - 4}$

Step 6.2

Simplify the result.

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Step 6.2.1

Add $7$ and $6$ .

$f (7) = \frac{\sqrt{13}}{7 - 4}$

Step 6.2.2

Subtract $4$ from $7$ .

$f (7) = \frac{\sqrt{13}}{3}$

Step 6.2.3

The final answer is $\frac{\sqrt{13}}{3}$ .

$\frac{\sqrt{13}}{3}$

Step 6.3

The $y$ value at $x = 7$ is $\frac{\sqrt{13}}{3}$ .

$y = \frac{\sqrt{13}}{3}$

Step 7

List the points to graph.

$(3, - 3), (- 5, - 0.\overline{1}), (- 4, - 0.17677669), (5, 3.31662479), (6, 1.7320508), (7, 1.20185042)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 5 & - 0.111 \\ - 4 & - 0.177 \\ 3 & - 3 \\ 5 & 3.317 \\ 6 & 1.732 \\ 7 & 1.202 \end{array}$

Step 9