Finite Math Examples

$h (x) = {(3 x - 3)}^{7}$

Step 1

Write $h (x) = {(3 x - 3)}^{7}$ as an equation.

$y = {(3 x - 3)}^{7}$

Step 2

Interchange the variables.

$x = {(3 y - 3)}^{7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as ${(3 y - 3)}^{7} = x$ .

${(3 y - 3)}^{7} = x$

Step 3.2

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

$3 y - 3 = \sqrt[7]{x}$

Step 3.3

Add $3$ to both sides of the equation.

$3 y = \sqrt[7]{x} + 3$

Step 3.4

Divide each term in $3 y = \sqrt[7]{x} + 3$ by $3$ and simplify.

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Step 3.4.1

Divide each term in $3 y = \sqrt[7]{x} + 3$ by $3$ .

$\frac{3 y}{3} = \frac{\sqrt[7]{x}}{3} + \frac{3}{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{\sqrt[7]{x}}{3} + \frac{3}{3}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\sqrt[7]{x}}{3} + \frac{3}{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Divide $3$ by $3$ .

$y = \frac{\sqrt[7]{x}}{3} + 1$

Step 4

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

$h^{- 1} (x) = \frac{\sqrt[7]{x}}{3} + 1$

Step 5

Verify if $h^{- 1} (x) = \frac{\sqrt[7]{x}}{3} + 1$ is the inverse of $h (x) = {(3 x - 3)}^{7}$ .

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Step 5.1

To verify the inverse, check if $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ .

Step 5.2

Evaluate $h^{- 1} (h (x))$ .

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Step 5.2.1

Set up the composite result function.

$h^{- 1} (h (x))$

Step 5.2.2

Evaluate $h^{- 1} ({(3 x - 3)}^{7})$ by substituting in the value of $h$ into $h^{- 1}$ .

$h^{- 1} ({(3 x - 3)}^{7}) = \frac{\sqrt[7]{{(3 x - 3)}^{7}}}{3} + 1$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

$h^{- 1} ({(3 x - 3)}^{7}) = \frac{3 x - 3}{3} + 1$

Step 5.2.3.1.2

Factor $3$ out of $3 x - 3$ .

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Step 5.2.3.1.2.1

Factor $3$ out of $3 x$ .

$h^{- 1} ({(3 x - 3)}^{7}) = \frac{3 (x) - 3}{3} + 1$

Step 5.2.3.1.2.2

Factor $3$ out of $- 3$ .

$h^{- 1} ({(3 x - 3)}^{7}) = \frac{3 (x) + 3 (- 1)}{3} + 1$

Step 5.2.3.1.2.3

Factor $3$ out of $3 (x) + 3 (- 1)$ .

$h^{- 1} ({(3 x - 3)}^{7}) = \frac{3 (x - 1)}{3} + 1$

Step 5.2.3.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.1

Cancel the common factor.

$h^{- 1} ({(3 x - 3)}^{7}) = \frac{3 (x - 1)}{3} + 1$

Step 5.2.3.2.2

Divide $x - 1$ by $1$ .

$h^{- 1} ({(3 x - 3)}^{7}) = x - 1 + 1$

Step 5.2.4

Combine the opposite terms in $x - 1 + 1$ .

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Step 5.2.4.1

Add $- 1$ and $1$ .

$h^{- 1} ({(3 x - 3)}^{7}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$h^{- 1} ({(3 x - 3)}^{7}) = x$

Step 5.3

Evaluate $h (h^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$h (h^{- 1} (x))$

Step 5.3.2

Evaluate $h (\frac{\sqrt[7]{x}}{3} + 1)$ by substituting in the value of $h^{- 1}$ into $h$ .

$h (\frac{\sqrt[7]{x}}{3} + 1) = {(3 (\frac{\sqrt[7]{x}}{3} + 1) - 3)}^{7}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$h (\frac{\sqrt[7]{x}}{3} + 1) = {(3 (\frac{\sqrt[7]{x}}{3}) + 3 \cdot 1 - 3)}^{7}$

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Cancel the common factor.

$h (\frac{\sqrt[7]{x}}{3} + 1) = {(3 (\frac{\sqrt[7]{x}}{3}) + 3 \cdot 1 - 3)}^{7}$

Step 5.3.3.2.2

Rewrite the expression.

$h (\frac{\sqrt[7]{x}}{3} + 1) = {(\sqrt[7]{x} + 3 \cdot 1 - 3)}^{7}$

Step 5.3.3.3

Multiply $3$ by $1$ .

$h (\frac{\sqrt[7]{x}}{3} + 1) = {(\sqrt[7]{x} + 3 - 3)}^{7}$

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Combine the opposite terms in $\sqrt[7]{x} + 3 - 3$ .

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Step 5.3.4.1.1

Subtract $3$ from $3$ .

$h (\frac{\sqrt[7]{x}}{3} + 1) = {(\sqrt[7]{x} + 0)}^{7}$

Step 5.3.4.1.2

Add $\sqrt[7]{x}$ and $0$ .

$h (\frac{\sqrt[7]{x}}{3} + 1) = {\sqrt[7]{x}}^{7}$

Step 5.3.4.2

Rewrite ${\sqrt[7]{x}}^{7}$ as $x$ .

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Step 5.3.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x}$ as $x^{\frac{1}{7}}$ .

$h (\frac{\sqrt[7]{x}}{3} + 1) = {(x^{\frac{1}{7}})}^{7}$

Step 5.3.4.2.2

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

$h (\frac{\sqrt[7]{x}}{3} + 1) = x^{\frac{1}{7} \cdot 7}$

Step 5.3.4.2.3

Combine $\frac{1}{7}$ and $7$ .

$h (\frac{\sqrt[7]{x}}{3} + 1) = x^{\frac{7}{7}}$

Step 5.3.4.2.4

Cancel the common factor of $7$ .

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Step 5.3.4.2.4.1

Cancel the common factor.

$h (\frac{\sqrt[7]{x}}{3} + 1) = x^{\frac{7}{7}}$

Step 5.3.4.2.4.2

Rewrite the expression.

$h (\frac{\sqrt[7]{x}}{3} + 1) = x$

Step 5.3.4.2.5

Simplify.

$h (\frac{\sqrt[7]{x}}{3} + 1) = x$

Step 5.4

Since $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ , then $h^{- 1} (x) = \frac{\sqrt[7]{x}}{3} + 1$ is the inverse of $h (x) = {(3 x - 3)}^{7}$ .

$h^{- 1} (x) = \frac{\sqrt[7]{x}}{3} + 1$