Pre-Algebra Examples

$y = \ln (9 - x^{2})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$9 - x^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $9$ from both sides of the equation.

$- x^{2} = - 9$

Step 1.2.2

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

Step 1.2.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 9}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

Step 1.2.3

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

$x = \pm \sqrt{9}$

Step 1.2.4

Simplify $\pm \sqrt{9}$ .

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Step 1.2.4.1

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

$x = \pm \sqrt{3^{2}}$

Step 1.2.4.2

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

$x = \pm 3$

Step 1.2.5

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

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Step 1.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Step 1.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

Step 1.2.5.3

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

$x = 3, - 3$

Step 1.3

The vertical asymptote occurs at $x = 3, x = - 3$ .

Vertical Asymptote: $x = 3, x = - 3$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = \ln (9 - {(0)}^{2})$

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

Raising $0$ to any positive power yields $0$ .

$f (0) = \ln (9 - 0)$

Step 2.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = \ln (9 + 0)$

Step 2.2.2

Add $9$ and $0$ .

$f (0) = \ln (9)$

Step 2.2.3

The final answer is $\ln (9)$ .

$\ln (9)$

Step 2.3

Convert $\ln (9)$ to decimal.

$y = 2.19722457$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

$f (1) = \ln (9 - {(1)}^{2})$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

One to any power is one.

$f (1) = \ln (9 - 1 \cdot 1)$

Step 3.2.1.2

Multiply $- 1$ by $1$ .

$f (1) = \ln (9 - 1)$

Step 3.2.2

Subtract $1$ from $9$ .

$f (1) = \ln (8)$

Step 3.2.3

The final answer is $\ln (8)$ .

$\ln (8)$

Step 3.3

Convert $\ln (8)$ to decimal.

$y = 2.07944154$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = \ln (9 - {(2)}^{2})$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f (2) = \ln (9 - 1 \cdot 4)$

Step 4.2.1.2

Multiply $- 1$ by $4$ .

$f (2) = \ln (9 - 4)$

Step 4.2.2

Subtract $4$ from $9$ .

$f (2) = \ln (5)$

Step 4.2.3

The final answer is $\ln (5)$ .

$\ln (5)$

Step 4.3

Convert $\ln (5)$ to decimal.

$y = 1.60943791$

Step 5

The log function can be graphed using the vertical asymptote at $x = 3, x = - 3$ and the points $(0, 2.19722457), (1, 2.07944154), (2, 1.60943791)$ .

Vertical Asymptote: $x = 3, x = - 3$

$\begin{array}{c} x & y \\ 0 & 2.197 \\ 1 & 2.079 \\ 2 & 1.609 \end{array}$

Step 6