Pre-Algebra Examples

$f (x) = \frac{\sqrt{1 + x^{2}}}{x}$

Step 1

Find the $y$ value at $x = - 1$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Move the negative one from the denominator of $\frac{\sqrt{1 + {(- 1)}^{2}}}{- 1}$ .

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

Step 1.2.2

Rewrite $- 1 \cdot \sqrt{1 + {(- 1)}^{2}}$ as $- \sqrt{1 + {(- 1)}^{2}}$ .

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

Step 1.2.3

Raise $- 1$ to the power of $2$ .

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

Step 1.2.4

Add $1$ and $1$ .

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

Step 1.2.5

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

$- \sqrt{2}$

Step 1.3

The $y$ value at $x = - 1$ is $- \sqrt{2}$ .

$y = - \sqrt{2}$

Step 2

Find the $y$ value at $x = - 2$ .

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Step 2.1

Replace the variable $x$ with $- 2$ in the expression.

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

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

Raise $- 2$ to the power of $2$ .

$f (- 2) = \frac{\sqrt{1 + 4}}{- 2}$

Step 2.2.1.2

Add $1$ and $4$ .

$f (- 2) = \frac{\sqrt{5}}{- 2}$

Step 2.2.2

Move the negative in front of the fraction.

$f (- 2) = - \frac{\sqrt{5}}{2}$

Step 2.2.3

The final answer is $- \frac{\sqrt{5}}{2}$ .

$- \frac{\sqrt{5}}{2}$

Step 2.3

The $y$ value at $x = - 2$ is $- \frac{\sqrt{5}}{2}$ .

$y = - \frac{\sqrt{5}}{2}$

Step 3

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

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Step 3.1

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

$f (- 3) = \frac{\sqrt{1 + {(- 3)}^{2}}}{- 3}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Raise $- 3$ to the power of $2$ .

$f (- 3) = \frac{\sqrt{1 + 9}}{- 3}$

Step 3.2.1.2

Add $1$ and $9$ .

$f (- 3) = \frac{\sqrt{10}}{- 3}$

Step 3.2.2

Move the negative in front of the fraction.

$f (- 3) = - \frac{\sqrt{10}}{3}$

Step 3.2.3

The final answer is $- \frac{\sqrt{10}}{3}$ .

$- \frac{\sqrt{10}}{3}$

Step 3.3

The $y$ value at $x = - 3$ is $- \frac{\sqrt{10}}{3}$ .

$y = - \frac{\sqrt{10}}{3}$

Step 4

Find the $y$ value at $x = 1$ .

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Step 4.1

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

$f (1) = \frac{\sqrt{1 + {(1)}^{2}}}{1}$

Step 4.2

Simplify the result.

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Step 4.2.1

Divide $\sqrt{1 + {(1)}^{2}}$ by $1$ .

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

Step 4.2.2

One to any power is one.

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

Step 4.2.3

Add $1$ and $1$ .

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

Step 4.2.4

The final answer is $\sqrt{2}$ .

$\sqrt{2}$

Step 4.3

The $y$ value at $x = 1$ is $\sqrt{2}$ .

$y = \sqrt{2}$

Step 5

Find the $y$ value at $x = 2$ .

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Step 5.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{\sqrt{1 + {(2)}^{2}}}{2}$

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

Raise $2$ to the power of $2$ .

$f (2) = \frac{\sqrt{1 + 4}}{2}$

Step 5.2.1.2

Add $1$ and $4$ .

$f (2) = \frac{\sqrt{5}}{2}$

Step 5.2.2

The final answer is $\frac{\sqrt{5}}{2}$ .

$\frac{\sqrt{5}}{2}$

Step 5.3

The $y$ value at $x = 2$ is $\frac{\sqrt{5}}{2}$ .

$y = \frac{\sqrt{5}}{2}$

Step 6

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

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Step 6.1

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

$f (3) = \frac{\sqrt{1 + {(3)}^{2}}}{3}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Raise $3$ to the power of $2$ .

$f (3) = \frac{\sqrt{1 + 9}}{3}$

Step 6.2.1.2

Add $1$ and $9$ .

$f (3) = \frac{\sqrt{10}}{3}$

Step 6.2.2

The final answer is $\frac{\sqrt{10}}{3}$ .

$\frac{\sqrt{10}}{3}$

Step 6.3

The $y$ value at $x = 3$ is $\frac{\sqrt{10}}{3}$ .

$y = \frac{\sqrt{10}}{3}$

Step 7

List the points to graph.

$(- 1, - 1.41421356), (- 2, - 1.11803398), (- 3, - 1.05409255), (1, 1.41421356), (2, 1.11803398), (3, 1.05409255)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 3 & - 1.054 \\ - 2 & - 1.118 \\ - 1 & - 1.414 \\ 1 & 1.414 \\ 2 & 1.118 \\ 3 & 1.054 \end{array}$

Step 9