Finite Math Examples

$f (x) = \frac{x^{5} + 2}{7}$

Step 1

Write $f (x) = \frac{x^{5} + 2}{7}$ as an equation.

$y = \frac{x^{5} + 2}{7}$

Step 2

Interchange the variables.

$x = \frac{y^{5} + 2}{7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y^{5} + 2}{7} = x$ .

$\frac{y^{5} + 2}{7} = x$

Step 3.2

Multiply both sides of the equation by $7$ .

$7 \frac{y^{5} + 2}{7} = 7 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $7$ .

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Step 3.3.1.1

Cancel the common factor.

$7 \frac{y^{5} + 2}{7} = 7 x$

Step 3.3.1.2

Rewrite the expression.

$y^{5} + 2 = 7 x$

Step 3.4

Subtract $2$ from both sides of the equation.

$y^{5} = 7 x - 2$

Step 3.5

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

$y = \sqrt[5]{7 x - 2}$

Step 4

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

$f^{- 1} (x) = \sqrt[5]{7 x - 2}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[5]{7 x - 2}$ is the inverse of $f (x) = \frac{x^{5} + 2}{7}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{x^{5} + 2}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x^{5} + 2}{7}) = \sqrt[5]{7 (\frac{x^{5} + 2}{7}) - 2}$

Step 5.2.3

Cancel the common factor of $7$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (\frac{x^{5} + 2}{7}) = \sqrt[5]{7 (\frac{x^{5} + 2}{7}) - 2}$

Step 5.2.3.2

Rewrite the expression.

$f^{- 1} (\frac{x^{5} + 2}{7}) = \sqrt[5]{x^{5} + 2 - 2}$

Step 5.2.4

Simplify by subtracting numbers.

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Step 5.2.4.1

Subtract $2$ from $2$ .

$f^{- 1} (\frac{x^{5} + 2}{7}) = \sqrt[5]{x^{5} + 0}$

Step 5.2.4.2

Add $x^{5}$ and $0$ .

$f^{- 1} (\frac{x^{5} + 2}{7}) = \sqrt[5]{x^{5}}$

Step 5.2.5

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

$f^{- 1} (\frac{x^{5} + 2}{7}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\sqrt[5]{7 x - 2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\sqrt[5]{7 x - 2}) = \frac{{(\sqrt[5]{7 x - 2})}^{5} + 2}{7}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{7 x - 2}$ as ${(7 x - 2)}^{\frac{1}{5}}$ .

$f (\sqrt[5]{7 x - 2}) = \frac{{({(7 x - 2)}^{\frac{1}{5}})}^{5} + 2}{7}$

Step 5.3.3.2

Multiply the exponents in ${({(7 x - 2)}^{\frac{1}{5}})}^{5}$ .

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Step 5.3.3.2.1

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

$f (\sqrt[5]{7 x - 2}) = \frac{{(7 x - 2)}^{\frac{1}{5} \cdot 5} + 2}{7}$

Step 5.3.3.2.2

Cancel the common factor of $5$ .

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Step 5.3.3.2.2.1

Cancel the common factor.

$f (\sqrt[5]{7 x - 2}) = \frac{{(7 x - 2)}^{\frac{1}{5} \cdot 5} + 2}{7}$

Step 5.3.3.2.2.2

Rewrite the expression.

$f (\sqrt[5]{7 x - 2}) = \frac{(7 x - 2) + 2}{7}$

Step 5.3.3.3

Simplify.

$f (\sqrt[5]{7 x - 2}) = \frac{7 x - 2 + 2}{7}$

Step 5.3.3.4

Add $- 2$ and $2$ .

$f (\sqrt[5]{7 x - 2}) = \frac{7 x + 0}{7}$

Step 5.3.3.5

Add $7 x$ and $0$ .

$f (\sqrt[5]{7 x - 2}) = \frac{7 x}{7}$

Step 5.3.4

Cancel the common factor of $7$ .

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Step 5.3.4.1

Cancel the common factor.

$f (\sqrt[5]{7 x - 2}) = \frac{7 x}{7}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (\sqrt[5]{7 x - 2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \sqrt[5]{7 x - 2}$ is the inverse of $f (x) = \frac{x^{5} + 2}{7}$ .

$f^{- 1} (x) = \sqrt[5]{7 x - 2}$