Algebra Examples

$f (x) = {(3 x)}^{- \frac{2}{3}}$ on the domain $x > 0$

Step 1

Find the range of the given function.

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Step 1.1

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

$(- \infty, \infty)$

Step 1.2

Convert $(- \infty, \infty)$ to an inequality.

$- \infty < y < \infty$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

$x = {(3 y)}^{- \frac{2}{3}}$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as ${(3 y)}^{- \frac{2}{3}} = x$ .

${(3 y)}^{- \frac{2}{3}} = x$

Step 2.2.2

Raise each side of the equation to the power of $- \frac{3}{2}$ to eliminate the fractional exponent on the left side.

${({(3 y)}^{- \frac{2}{3}})}^{- \frac{3}{2}} = \pm x^{- \frac{3}{2}}$

Step 2.2.3

Simplify the exponent.

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Step 2.2.3.1

Simplify the left side.

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Step 2.2.3.1.1

Simplify ${({(3 y)}^{- \frac{2}{3}})}^{- \frac{3}{2}}$ .

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Step 2.2.3.1.1.1

Multiply the exponents in ${({(3 y)}^{- \frac{2}{3}})}^{- \frac{3}{2}}$ .

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Step 2.2.3.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(3 y)}^{- \frac{2}{3} (- \frac{3}{2})} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.3.1.1.1.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

${(3 y)}^{\frac{- 2}{3} (- \frac{3}{2})} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.2.2

Move the leading negative in $- \frac{3}{2}$ into the numerator.

${(3 y)}^{\frac{- 2}{3} \cdot \frac{- 3}{2}} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.2.3

Factor $2$ out of $- 2$ .

${(3 y)}^{\frac{2 (- 1)}{3} \cdot \frac{- 3}{2}} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.2.4

Cancel the common factor.

${(3 y)}^{\frac{2 \cdot - 1}{3} \cdot \frac{- 3}{2}} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.2.5

Rewrite the expression.

${(3 y)}^{\frac{- 1}{3} \cdot - 3} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.2.3.1.1.1.3.1

Factor $3$ out of $- 3$ .

${(3 y)}^{\frac{- 1}{3} \cdot (3 (- 1))} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.3.2

Cancel the common factor.

${(3 y)}^{\frac{- 1}{3} \cdot (3 \cdot - 1)} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.3.3

Rewrite the expression.

${(3 y)}^{- - 1} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.1.4

Multiply $- 1$ by $- 1$ .

${(3 y)}^{1} = \pm x^{- \frac{3}{2}}$

Step 2.2.3.1.1.2

Simplify.

$3 y = \pm x^{- \frac{3}{2}}$

Step 2.2.3.2

Simplify the right side.

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Step 2.2.3.2.1

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

$3 y = \pm \frac{1}{x^{\frac{3}{2}}}$

Step 2.2.4

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

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Step 2.2.4.1

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

$3 y = \frac{1}{x^{\frac{3}{2}}}$

Step 2.2.4.2

Divide each term in $3 y = \frac{1}{x^{\frac{3}{2}}}$ by $3$ and simplify.

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Step 2.2.4.2.1

Divide each term in $3 y = \frac{1}{x^{\frac{3}{2}}}$ by $3$ .

$\frac{3 y}{3} = \frac{\frac{1}{x^{\frac{3}{2}}}}{3}$

Step 2.2.4.2.2

Simplify the left side.

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Step 2.2.4.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.4.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\frac{1}{x^{\frac{3}{2}}}}{3}$

Step 2.2.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{x^{\frac{3}{2}}}}{3}$

Step 2.2.4.2.3

Simplify the right side.

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Step 2.2.4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{1}{x^{\frac{3}{2}}} \cdot \frac{1}{3}$

Step 2.2.4.2.3.2

Combine.

$y = \frac{1 \cdot 1}{x^{\frac{3}{2}} \cdot 3}$

Step 2.2.4.2.3.3

Simplify the expression.

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Step 2.2.4.2.3.3.1

Multiply $1$ by $1$ .

$y = \frac{1}{x^{\frac{3}{2}} \cdot 3}$

Step 2.2.4.2.3.3.2

Move $3$ to the left of $x^{\frac{3}{2}}$ .

$y = \frac{1}{3 x^{\frac{3}{2}}}$

Step 2.2.4.3

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

$3 y = - \frac{1}{x^{\frac{3}{2}}}$

Step 2.2.4.4

Divide each term in $3 y = - \frac{1}{x^{\frac{3}{2}}}$ by $3$ and simplify.

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Step 2.2.4.4.1

Divide each term in $3 y = - \frac{1}{x^{\frac{3}{2}}}$ by $3$ .

$\frac{3 y}{3} = \frac{- \frac{1}{x^{\frac{3}{2}}}}{3}$

Step 2.2.4.4.2

Simplify the left side.

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Step 2.2.4.4.2.1

Cancel the common factor of $3$ .

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Step 2.2.4.4.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- \frac{1}{x^{\frac{3}{2}}}}{3}$

Step 2.2.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{1}{x^{\frac{3}{2}}}}{3}$

Step 2.2.4.4.3

Simplify the right side.

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Step 2.2.4.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{1}{x^{\frac{3}{2}}} \cdot \frac{1}{3}$

Step 2.2.4.4.3.2

Multiply $\frac{1}{3}$ by $\frac{1}{x^{\frac{3}{2}}}$ .

$y = - \frac{1}{3 x^{\frac{3}{2}}}$

Step 2.2.4.5

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

$y = \frac{1}{3 x^{\frac{3}{2}}}$

$y = - \frac{1}{3 x^{\frac{3}{2}}}$

$y = \frac{1}{3 x^{\frac{3}{2}}}$

$y = - \frac{1}{3 x^{\frac{3}{2}}}$

$y = \frac{1}{3 x^{\frac{3}{2}}}$

$y = - \frac{1}{3 x^{\frac{3}{2}}}$

Step 2.3

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

$f^{- 1} (x) = \frac{1}{3 x^{\frac{3}{2}}}, - \frac{1}{3 x^{\frac{3}{2}}}$

Step 3

Find the inverse using the domain and the range of the original function.

$f^{- 1} (x) = \frac{1}{3 x^{\frac{3}{2}}}, - \infty < x < \infty$

Step 4