Finite Math Examples

$f (x) = \frac{- 5}{x^{2 - 4}}$

Step 1

Write $f (x) = \frac{- 5}{x^{2 - 4}}$ as an equation.

$y = \frac{- 5}{x^{2 - 4}}$

Step 2

Interchange the variables.

$x = \frac{- 5}{y^{2 - 4}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{- 5}{y^{2 - 4}} = x$ .

$\frac{- 5}{y^{2 - 4}} = x$

Step 3.2

Factor each term.

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Step 3.2.1

Move $y^{2 - 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$- 5 y^{- (2 - 4)} = x$

Step 3.2.2

Subtract $4$ from $2$ .

$- 5 y^{- - 2} = x$

Step 3.2.3

Multiply $- 1$ by $- 2$ .

$- 5 y^{2} = x$

Step 3.3

Solve the equation.

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Step 3.3.1

Divide each term in $- 5 y^{2} = x$ by $- 5$ and simplify.

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Step 3.3.1.1

Divide each term in $- 5 y^{2} = x$ by $- 5$ .

$\frac{- 5 y^{2}}{- 5} = \frac{x}{- 5}$

Step 3.3.1.2

Simplify the left side.

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Step 3.3.1.2.1

Cancel the common factor of $- 5$ .

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Step 3.3.1.2.1.1

Cancel the common factor.

$\frac{- 5 y^{2}}{- 5} = \frac{x}{- 5}$

Step 3.3.1.2.1.2

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

$y^{2} = \frac{x}{- 5}$

Step 3.3.1.3

Simplify the right side.

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Step 3.3.1.3.1

Move the negative in front of the fraction.

$y^{2} = - \frac{x}{5}$

Step 3.3.2

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

$y = \pm \sqrt{- \frac{x}{5}}$

Step 3.3.3

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

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Step 3.3.3.1

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

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

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

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

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

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

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

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

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

Step 4

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

$f^{- 1} (x) = \sqrt{- \frac{x}{5}}, - \sqrt{- \frac{x}{5}}$

Step 5

Verify if $f^{- 1} (x) = \sqrt{- \frac{x}{5}}, - \sqrt{- \frac{x}{5}}$ is the inverse of $f (x) = \frac{- 5}{x^{2 - 4}}$ .

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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 $f (x) = \frac{- 5}{x^{2 - 4}}$ and $f^{- 1} (x) = \sqrt{- \frac{x}{5}}, - \sqrt{- \frac{x}{5}}$ and compare them.

Step 5.2

Find the range of $f (x) = \frac{- 5}{x^{2 - 4}}$ .

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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:

$(- \infty, 0]$

Step 5.3

Find the domain of $\sqrt{- \frac{x}{5}}$ .

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Step 5.3.1

Set the radicand in $\sqrt{- \frac{x}{5}}$ greater than or equal to $0$ to find where the expression is defined.

$- \frac{x}{5} \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Divide each term in $- \frac{x}{5} \geq 0$ by $- 1$ and simplify.

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Step 5.3.2.1.1

Divide each term in $- \frac{x}{5} \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{5}}{- 1} \leq \frac{0}{- 1}$

Step 5.3.2.1.2

Simplify the left side.

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Step 5.3.2.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{5}}{1} \leq \frac{0}{- 1}$

Step 5.3.2.1.2.2

Divide $\frac{x}{5}$ by $1$ .

$\frac{x}{5} \leq \frac{0}{- 1}$

Step 5.3.2.1.3

Simplify the right side.

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Step 5.3.2.1.3.1

Divide $0$ by $- 1$ .

$\frac{x}{5} \leq 0$

Step 5.3.2.2

Multiply both sides by $5$ .

$\frac{x}{5} \cdot 5 \leq 0 \cdot 5$

Step 5.3.2.3

Simplify.

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Step 5.3.2.3.1

Simplify the left side.

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Step 5.3.2.3.1.1

Cancel the common factor of $5$ .

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Step 5.3.2.3.1.1.1

Cancel the common factor.

$\frac{x}{5} \cdot 5 \leq 0 \cdot 5$

Step 5.3.2.3.1.1.2

Rewrite the expression.

$x \leq 0 \cdot 5$

Step 5.3.2.3.2

Simplify the right side.

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Step 5.3.2.3.2.1

Multiply $0$ by $5$ .

$x \leq 0$

Step 5.3.3

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

$(- \infty, 0]$

Step 5.4

Find the domain of $f (x) = \frac{- 5}{x^{2 - 4}}$ .

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Step 5.4.1

Set the denominator in $\frac{- 5}{x^{2 - 4}}$ equal to $0$ to find where the expression is undefined.

$x^{2 - 4} = 0$

Step 5.4.2

Solve for $x$ .

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Step 5.4.2.1

Simplify $x^{2 - 4}$ .

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Step 5.4.2.1.1

Subtract $4$ from $2$ .

$x^{- 2} = 0$

Step 5.4.2.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{2}} = 0$

Step 5.4.2.2

Set the numerator equal to zero.

$1 = 0$

Step 5.4.2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 5.4.3

Set the base in $x^{2 - 4}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5.4.4

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

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

Step 5.5

Find the range of the inverse.

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Step 5.5.1

Find the range of $\sqrt{- \frac{x}{5}}$ .

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Step 5.5.1.1

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

Interval Notation:

$[0, \infty)$

Step 5.5.2

Find the range of $- \sqrt{- \frac{x}{5}}$ .

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Step 5.5.2.1

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

Interval Notation:

$(- \infty, 0]$

Step 5.5.3

Find the union of $[0, \infty), (- \infty, 0]$ .

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Step 5.5.3.1

The union consists of all of the elements that are contained in each interval.

$(- \infty, \infty)$

Step 5.6

Since the range of $f^{- 1} (x) = \sqrt{- \frac{x}{5}}, - \sqrt{- \frac{x}{5}}$ is not equal to the domain of $f (x) = \frac{- 5}{x^{2 - 4}}$ , then $f^{- 1} (x) = \sqrt{- \frac{x}{5}}, - \sqrt{- \frac{x}{5}}$ is not an inverse of $f (x) = \frac{- 5}{x^{2 - 4}}$ .

There is no inverse

Step 6