Finite Math Examples

$f (x) = \frac{8 x}{x^{2} - 64}$

Step 1

Write $f (x) = \frac{8 x}{x^{2} - 64}$ as an equation.

$y = \frac{8 x}{x^{2} - 64}$

Step 2

Interchange the variables.

$x = \frac{8 y}{y^{2} - 64}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $y^{2} - 64$ .

$(y^{2} - 64) (x) = (\frac{8 y}{y^{2} - 64}) (y^{2} - 64)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$y^{2} x - 64 x = (\frac{8 y}{y^{2} - 64}) (y^{2} - 64)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $(\frac{8 y}{y^{2} - 64}) (y^{2} - 64)$ .

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Step 3.3.1.1

Simplify the denominator.

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Step 3.3.1.1.1

Rewrite $64$ as $8^{2}$ .

$y^{2} x - 64 x = \frac{8 y}{y^{2} - 8^{2}} (y^{2} - 64)$

Step 3.3.1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 8$ .

$y^{2} x - 64 x = \frac{8 y}{(y + 8) (y - 8)} (y^{2} - 64)$

Step 3.3.1.2

Multiply $\frac{8 y}{(y + 8) (y - 8)}$ by $y^{2} - 64$ .

$y^{2} x - 64 x = \frac{8 y (y^{2} - 64)}{(y + 8) (y - 8)}$

Step 3.3.1.3

Simplify the numerator.

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Step 3.3.1.3.1

Rewrite $64$ as $8^{2}$ .

$y^{2} x - 64 x = \frac{8 y (y^{2} - 8^{2})}{(y + 8) (y - 8)}$

Step 3.3.1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 8$ .

$y^{2} x - 64 x = \frac{8 y (y + 8) (y - 8)}{(y + 8) (y - 8)}$

Step 3.3.1.4

Reduce the expression by cancelling the common factors.

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Step 3.3.1.4.1

Cancel the common factor of $y + 8$ .

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Step 3.3.1.4.1.1

Cancel the common factor.

$y^{2} x - 64 x = \frac{8 y (y + 8) (y - 8)}{(y + 8) (y - 8)}$

Step 3.3.1.4.1.2

Rewrite the expression.

$y^{2} x - 64 x = \frac{8 y (y - 8)}{y - 8}$

Step 3.3.1.4.2

Cancel the common factor of $y - 8$ .

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Step 3.3.1.4.2.1

Cancel the common factor.

$y^{2} x - 64 x = \frac{8 y (y - 8)}{y - 8}$

Step 3.3.1.4.2.2

Divide $8 y$ by $1$ .

$y^{2} x - 64 x = 8 y$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $8 y$ from both sides of the equation.

$y^{2} x - 64 x - 8 y = 0$

Step 3.4.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4.3

Substitute the values $a = x$ , $b = - 8$ , and $c = - 64 x$ into the quadratic formula and solve for $y$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (x \cdot (- 64 x))}}{2 x}$

Step 3.4.4

Simplify.

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Step 3.4.4.1

Simplify the numerator.

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Step 3.4.4.1.1

Raise $- 8$ to the power of $2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot x \cdot (- 64 x)}}{2 x}$

Step 3.4.4.1.2

Rewrite using the commutative property of multiplication.

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 x \cdot x)}}{2 x}$

Step 3.4.4.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.4.1.3.1

Move $x$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 (x \cdot x))}}{2 x}$

Step 3.4.4.1.3.2

Multiply $x$ by $x$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 x^{2})}}{2 x}$

Step 3.4.4.1.4

Multiply $- 4$ by $- 64$ .

$y = \frac{8 \pm \sqrt{64 + 256 x^{2}}}{2 x}$

Step 3.4.4.1.5

Factor $64$ out of $64 + 256 x^{2}$ .

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Step 3.4.4.1.5.1

Factor $64$ out of $64$ .

$y = \frac{8 \pm \sqrt{64 (1) + 256 x^{2}}}{2 x}$

Step 3.4.4.1.5.2

Factor $64$ out of $256 x^{2}$ .

$y = \frac{8 \pm \sqrt{64 (1) + 64 (4 x^{2})}}{2 x}$

Step 3.4.4.1.5.3

Factor $64$ out of $64 (1) + 64 (4 x^{2})$ .

$y = \frac{8 \pm \sqrt{64 (1 + 4 x^{2})}}{2 x}$

Step 3.4.4.1.6

Rewrite $64 (1 + 4 x^{2})$ as $8^{2} (1^{2} + 4 x^{2})$ .

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Step 3.4.4.1.6.1

Rewrite $64$ as $8^{2}$ .

$y = \frac{8 \pm \sqrt{8^{2} (1 + 4 x^{2})}}{2 x}$

Step 3.4.4.1.6.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{8 \pm \sqrt{8^{2} (1^{2} + 4 x^{2})}}{2 x}$

Step 3.4.4.1.7

Pull terms out from under the radical.

$y = \frac{8 \pm 8 \sqrt{1^{2} + 4 x^{2}}}{2 x}$

Step 3.4.4.1.8

One to any power is one.

$y = \frac{8 \pm 8 \sqrt{1 + 4 x^{2}}}{2 x}$

Step 3.4.4.2

Simplify $\frac{8 \pm 8 \sqrt{1 + 4 x^{2}}}{2 x}$ .

$y = \frac{4 \pm 4 \sqrt{1 + 4 x^{2}}}{x}$

Step 3.4.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.4.5.1

Simplify the numerator.

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Step 3.4.5.1.1

Raise $- 8$ to the power of $2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot x \cdot (- 64 x)}}{2 x}$

Step 3.4.5.1.2

Rewrite using the commutative property of multiplication.

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 x \cdot x)}}{2 x}$

Step 3.4.5.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.5.1.3.1

Move $x$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 (x \cdot x))}}{2 x}$

Step 3.4.5.1.3.2

Multiply $x$ by $x$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 x^{2})}}{2 x}$

Step 3.4.5.1.4

Multiply $- 4$ by $- 64$ .

$y = \frac{8 \pm \sqrt{64 + 256 x^{2}}}{2 x}$

Step 3.4.5.1.5

Factor $64$ out of $64 + 256 x^{2}$ .

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Step 3.4.5.1.5.1

Factor $64$ out of $64$ .

$y = \frac{8 \pm \sqrt{64 (1) + 256 x^{2}}}{2 x}$

Step 3.4.5.1.5.2

Factor $64$ out of $256 x^{2}$ .

$y = \frac{8 \pm \sqrt{64 (1) + 64 (4 x^{2})}}{2 x}$

Step 3.4.5.1.5.3

Factor $64$ out of $64 (1) + 64 (4 x^{2})$ .

$y = \frac{8 \pm \sqrt{64 (1 + 4 x^{2})}}{2 x}$

Step 3.4.5.1.6

Rewrite $64 (1 + 4 x^{2})$ as $8^{2} (1^{2} + 4 x^{2})$ .

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Step 3.4.5.1.6.1

Rewrite $64$ as $8^{2}$ .

$y = \frac{8 \pm \sqrt{8^{2} (1 + 4 x^{2})}}{2 x}$

Step 3.4.5.1.6.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{8 \pm \sqrt{8^{2} (1^{2} + 4 x^{2})}}{2 x}$

Step 3.4.5.1.7

Pull terms out from under the radical.

$y = \frac{8 \pm 8 \sqrt{1^{2} + 4 x^{2}}}{2 x}$

Step 3.4.5.1.8

One to any power is one.

$y = \frac{8 \pm 8 \sqrt{1 + 4 x^{2}}}{2 x}$

Step 3.4.5.2

Simplify $\frac{8 \pm 8 \sqrt{1 + 4 x^{2}}}{2 x}$ .

$y = \frac{4 \pm 4 \sqrt{1 + 4 x^{2}}}{x}$

Step 3.4.5.3

Change the $\pm$ to $+$ .

$y = \frac{4 + 4 \sqrt{1 + 4 x^{2}}}{x}$

Step 3.4.5.4

Factor $4$ out of $4 + 4 \sqrt{1 + 4 x^{2}}$ .

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Step 3.4.5.4.1

Factor $4$ out of $4$ .

$y = \frac{4 \cdot 1 + 4 \sqrt{1 + 4 x^{2}}}{x}$

Step 3.4.5.4.2

Factor $4$ out of $4 \cdot 1 + 4 \sqrt{1 + 4 x^{2}}$ .

$y = \frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}$

Step 3.4.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.4.6.1

Simplify the numerator.

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Step 3.4.6.1.1

Raise $- 8$ to the power of $2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot x \cdot (- 64 x)}}{2 x}$

Step 3.4.6.1.2

Rewrite using the commutative property of multiplication.

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 x \cdot x)}}{2 x}$

Step 3.4.6.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.6.1.3.1

Move $x$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 (x \cdot x))}}{2 x}$

Step 3.4.6.1.3.2

Multiply $x$ by $x$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot (- 64 x^{2})}}{2 x}$

Step 3.4.6.1.4

Multiply $- 4$ by $- 64$ .

$y = \frac{8 \pm \sqrt{64 + 256 x^{2}}}{2 x}$

Step 3.4.6.1.5

Factor $64$ out of $64 + 256 x^{2}$ .

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Step 3.4.6.1.5.1

Factor $64$ out of $64$ .

$y = \frac{8 \pm \sqrt{64 (1) + 256 x^{2}}}{2 x}$

Step 3.4.6.1.5.2

Factor $64$ out of $256 x^{2}$ .

$y = \frac{8 \pm \sqrt{64 (1) + 64 (4 x^{2})}}{2 x}$

Step 3.4.6.1.5.3

Factor $64$ out of $64 (1) + 64 (4 x^{2})$ .

$y = \frac{8 \pm \sqrt{64 (1 + 4 x^{2})}}{2 x}$

Step 3.4.6.1.6

Rewrite $64 (1 + 4 x^{2})$ as $8^{2} (1^{2} + 4 x^{2})$ .

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Step 3.4.6.1.6.1

Rewrite $64$ as $8^{2}$ .

$y = \frac{8 \pm \sqrt{8^{2} (1 + 4 x^{2})}}{2 x}$

Step 3.4.6.1.6.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{8 \pm \sqrt{8^{2} (1^{2} + 4 x^{2})}}{2 x}$

Step 3.4.6.1.7

Pull terms out from under the radical.

$y = \frac{8 \pm 8 \sqrt{1^{2} + 4 x^{2}}}{2 x}$

Step 3.4.6.1.8

One to any power is one.

$y = \frac{8 \pm 8 \sqrt{1 + 4 x^{2}}}{2 x}$

Step 3.4.6.2

Simplify $\frac{8 \pm 8 \sqrt{1 + 4 x^{2}}}{2 x}$ .

$y = \frac{4 \pm 4 \sqrt{1 + 4 x^{2}}}{x}$

Step 3.4.6.3

Change the $\pm$ to $-$ .

$y = \frac{4 - 4 \sqrt{1 + 4 x^{2}}}{x}$

Step 3.4.6.4

Factor $4$ out of $4 - 4 \sqrt{1 + 4 x^{2}}$ .

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Step 3.4.6.4.1

Factor $4$ out of $4$ .

$y = \frac{4 (1) - 4 \sqrt{1 + 4 x^{2}}}{x}$

Step 3.4.6.4.2

Factor $4$ out of $- 4 \sqrt{1 + 4 x^{2}}$ .

$y = \frac{4 (1) + 4 (- \sqrt{1 + 4 x^{2}})}{x}$

Step 3.4.6.4.3

Factor $4$ out of $4 (1) + 4 (- \sqrt{1 + 4 x^{2}})$ .

$y = \frac{4 (1 - \sqrt{1 + 4 x^{2}})}{x}$

Step 3.4.7

The final answer is the combination of both solutions.

$y = \frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}$

$y = \frac{4 (1 - \sqrt{1 + 4 x^{2}})}{x}$

$y = \frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}$

$y = \frac{4 (1 - \sqrt{1 + 4 x^{2}})}{x}$

$y = \frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}$

$y = \frac{4 (1 - \sqrt{1 + 4 x^{2}})}{x}$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

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

Step 5

Verify if $f^{- 1} (x) = \frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}, \frac{4 (1 - \sqrt{1 + 4 x^{2}})}{x}$ is the inverse of $f (x) = \frac{8 x}{x^{2} - 64}$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \frac{8 x}{x^{2} - 64}$ and $f^{- 1} (x) = \frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}, \frac{4 (1 - \sqrt{1 + 4 x^{2}})}{x}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{8 x}{x^{2} - 64}$ .

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Step 5.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Step 5.3

Find the domain of $\frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}$ .

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Step 5.3.1

Set the radicand in $\sqrt{1 + 4 x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$1 + 4 x^{2} \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Subtract $1$ from both sides of the inequality.

$4 x^{2} \geq - 1$

Step 5.3.2.2

Divide each term in $4 x^{2} \geq - 1$ by $4$ and simplify.

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Step 5.3.2.2.1

Divide each term in $4 x^{2} \geq - 1$ by $4$ .

$\frac{4 x^{2}}{4} \geq \frac{- 1}{4}$

Step 5.3.2.2.2

Simplify the left side.

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Step 5.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 5.3.2.2.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} \geq \frac{- 1}{4}$

Step 5.3.2.2.2.1.2

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

$x^{2} \geq \frac{- 1}{4}$

Step 5.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.1

Move the negative in front of the fraction.

$x^{2} \geq - \frac{1}{4}$

Step 5.3.2.3

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 5.3.3

Set the denominator in $\frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5.3.4

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

$(- \infty, 0) \cup (0, \infty)$

Step 5.4

Since the domain of $f^{- 1} (x) = \frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}, \frac{4 (1 - \sqrt{1 + 4 x^{2}})}{x}$ is not equal to the range of $f (x) = \frac{8 x}{x^{2} - 64}$ , then $f^{- 1} (x) = \frac{4 (1 + \sqrt{1 + 4 x^{2}})}{x}, \frac{4 (1 - \sqrt{1 + 4 x^{2}})}{x}$ is not an inverse of $f (x) = \frac{8 x}{x^{2} - 64}$ .

There is no inverse

Step 6