Trigonometry Examples

$f (x) = - \frac{3}{8} x^{7}$

Step 1

Write $f (x) = - \frac{3}{8} x^{7}$ as an equation.

$y = - \frac{3}{8} x^{7}$

Step 2

Interchange the variables.

$x = - \frac{3}{8} y^{7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \frac{3}{8} y^{7} = x$ .

$- \frac{3}{8} y^{7} = x$

Step 3.2

Multiply both sides of the equation by $- \frac{8}{3}$ .

$- \frac{8}{3} (- \frac{3}{8} y^{7}) = - \frac{8}{3} x$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $- \frac{8}{3} (- \frac{3}{8} y^{7})$ .

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Step 3.3.1.1.1

Combine $y^{7}$ and $\frac{3}{8}$ .

$- \frac{8}{3} (- \frac{y^{7} \cdot 3}{8}) = - \frac{8}{3} x$

Step 3.3.1.1.2

Cancel the common factor of $8$ .

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Step 3.3.1.1.2.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$\frac{- 8}{3} (- \frac{y^{7} \cdot 3}{8}) = - \frac{8}{3} x$

Step 3.3.1.1.2.2

Move the leading negative in $- \frac{y^{7} \cdot 3}{8}$ into the numerator.

$\frac{- 8}{3} \cdot \frac{- y^{7} \cdot 3}{8} = - \frac{8}{3} x$

Step 3.3.1.1.2.3

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{3} \cdot \frac{- y^{7} \cdot 3}{8} = - \frac{8}{3} x$

Step 3.3.1.1.2.4

Cancel the common factor.

$\frac{8 \cdot - 1}{3} \cdot \frac{- y^{7} \cdot 3}{8} = - \frac{8}{3} x$

Step 3.3.1.1.2.5

Rewrite the expression.

$\frac{- 1}{3} (- y^{7} \cdot 3) = - \frac{8}{3} x$

Step 3.3.1.1.3

Cancel the common factor of $3$ .

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Step 3.3.1.1.3.1

Factor $3$ out of $- y^{7} \cdot 3$ .

$\frac{- 1}{3} (3 \cdot (- y^{7})) = - \frac{8}{3} x$

Step 3.3.1.1.3.2

Cancel the common factor.

$\frac{- 1}{3} (3 \cdot (- y^{7})) = - \frac{8}{3} x$

Step 3.3.1.1.3.3

Rewrite the expression.

$- - y^{7} = - \frac{8}{3} x$

Step 3.3.1.1.4

Multiply.

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Step 3.3.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 y^{7} = - \frac{8}{3} x$

Step 3.3.1.1.4.2

Multiply $y^{7}$ by $1$ .

$y^{7} = - \frac{8}{3} x$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \frac{8}{3} x$ .

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Step 3.3.2.1.1

Combine $x$ and $\frac{8}{3}$ .

$y^{7} = - \frac{x \cdot 8}{3}$

Step 3.3.2.1.2

Move $8$ to the left of $x$ .

$y^{7} = - \frac{8 x}{3}$

Step 3.4

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

$y = \sqrt[7]{- \frac{8 x}{3}}$

Step 3.5

Simplify $\sqrt[7]{- \frac{8 x}{3}}$ .

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Step 3.5.1

Rewrite $- \frac{8 x}{3}$ as ${({(- 1)}^{7})}^{7} \frac{8 x}{3}$ .

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Step 3.5.1.1

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$y = \sqrt[7]{{(- 1)}^{7} \frac{8 x}{3}}$

Step 3.5.1.2

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$y = \sqrt[7]{{({(- 1)}^{7})}^{7} \frac{8 x}{3}}$

Step 3.5.2

Pull terms out from under the radical.

$y = {(- 1)}^{7} \sqrt[7]{\frac{8 x}{3}}$

Step 3.5.3

Raise $- 1$ to the power of $7$ .

$y = - \sqrt[7]{\frac{8 x}{3}}$

Step 3.5.4

Rewrite $\sqrt[7]{\frac{8 x}{3}}$ as $\frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}$ .

$y = - \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}$

Step 4

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

$f^{- 1} (x) = - \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}$

Step 5

Verify if $f^{- 1} (x) = - \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}$ is the inverse of $f (x) = - \frac{3}{8} x^{7}$ .

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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 $f^{- 1} (- \frac{3}{8} x^{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{8 (- \frac{3}{8} x^{7})}}{\sqrt[7]{3}}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Multiply $- 1$ by $8$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{- 8 (\frac{3}{8} x^{7})}}{\sqrt[7]{3}}$

Step 5.2.3.2

Combine $\frac{3}{8}$ and $- 8$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{\frac{3 \cdot - 8}{8} x^{7}}}{\sqrt[7]{3}}$

Step 5.2.3.3

Combine $\frac{3 \cdot - 8}{8}$ and $x^{7}$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{\frac{3 \cdot (- 8 x^{7})}{8}}}{\sqrt[7]{3}}$

Step 5.2.3.4

Reduce the expression by cancelling the common factors.

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Step 5.2.3.4.1

Reduce the expression $\frac{3 \cdot - 8 x^{7}}{8}$ by cancelling the common factors.

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Step 5.2.3.4.1.1

Factor $8$ out of $3 \cdot - 8 x^{7}$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{\frac{8 (3 \cdot (- 1 x^{7}))}{8}}}{\sqrt[7]{3}}$

Step 5.2.3.4.1.2

Factor $8$ out of $8$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{\frac{8 (3 \cdot (- 1 x^{7}))}{8 (1)}}}{\sqrt[7]{3}}$

Step 5.2.3.4.1.3

Cancel the common factor.

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{\frac{8 (3 \cdot (- 1 x^{7}))}{8 \cdot 1}}}{\sqrt[7]{3}}$

Step 5.2.3.4.1.4

Rewrite the expression.

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{\frac{3 \cdot (- 1 x^{7})}{1}}}{\sqrt[7]{3}}$

Step 5.2.3.4.2

Divide $3 \cdot - 1 x^{7}$ by $1$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{3 \cdot (- 1 x^{7})}}{\sqrt[7]{3}}$

Step 5.2.3.5

Combine exponents.

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Step 5.2.3.5.1

Factor out negative.

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{- (3 x^{7})}}{\sqrt[7]{3}}$

Step 5.2.3.5.2

Multiply $3$ by $- 1$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{- 3 x^{7}}}{\sqrt[7]{3}}$

Step 5.2.3.6

Rewrite $- 3 x^{7}$ as ${(- x)}^{7} \cdot 3$ .

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Step 5.2.3.6.1

Rewrite $- 3$ as $- 1 (3)$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{- 1 \cdot 3 x^{7}}}{\sqrt[7]{3}}$

Step 5.2.3.6.2

Rewrite $- 1$ as ${(- 1)}^{7}$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{{(- 1)}^{7} \cdot (3 x^{7})}}{\sqrt[7]{3}}$

Step 5.2.3.6.3

Move $3$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{{(- 1)}^{7} x^{7} \cdot 3}}{\sqrt[7]{3}}$

Step 5.2.3.6.4

Rewrite ${(- 1)}^{7} x^{7}$ as ${(- x)}^{7}$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{\sqrt[7]{{(- x)}^{7} \cdot 3}}{\sqrt[7]{3}}$

Step 5.2.3.7

Pull terms out from under the radical.

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{- x \sqrt[7]{3}}{\sqrt[7]{3}}$

Step 5.2.4

Cancel the common factor of $\sqrt[7]{3}$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (- \frac{3}{8} x^{7}) = - \frac{- x \sqrt[7]{3}}{\sqrt[7]{3}}$

Step 5.2.4.2

Divide $- x$ by $1$ .

$f^{- 1} (- \frac{3}{8} x^{7}) = x$

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 $f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = - \frac{3}{8} \cdot {(- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}})}^{7}$

Step 5.3.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.3.3.1

Apply the product rule to $- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = - \frac{3}{8} \cdot ({(- 1)}^{7} {(\frac{\sqrt[7]{8 x}}{\sqrt[7]{3}})}^{7})$

Step 5.3.3.2

Apply the product rule to $\frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = - \frac{3}{8} \cdot ({(- 1)}^{7} (\frac{{\sqrt[7]{8 x}}^{7}}{{\sqrt[7]{3}}^{7}}))$

Step 5.3.4

Multiply $- 1$ by ${(- 1)}^{7}$ by adding the exponents.

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Step 5.3.4.1

Move ${(- 1)}^{7}$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{7} \cdot (- 1 (\frac{3}{8})) \cdot \frac{{\sqrt[7]{8 x}}^{7}}{{\sqrt[7]{3}}^{7}}$

Step 5.3.4.2

Multiply ${(- 1)}^{7}$ by $- 1$ .

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Step 5.3.4.2.1

Raise $- 1$ to the power of $1$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{7} \cdot ((- 1) (\frac{3}{8})) \cdot \frac{{\sqrt[7]{8 x}}^{7}}{{\sqrt[7]{3}}^{7}}$

Step 5.3.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{7 + 1} (\frac{3}{8}) \cdot \frac{{\sqrt[7]{8 x}}^{7}}{{\sqrt[7]{3}}^{7}}$

Step 5.3.4.3

Add $7$ and $1$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{{\sqrt[7]{8 x}}^{7}}{{\sqrt[7]{3}}^{7}}$

Step 5.3.5

Rewrite ${\sqrt[7]{8 x}}^{7}$ as $8 x$ .

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Step 5.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{8 x}$ as ${(8 x)}^{\frac{1}{7}}$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{{({(8 x)}^{\frac{1}{7}})}^{7}}{{\sqrt[7]{3}}^{7}}$

Step 5.3.5.2

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

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{{(8 x)}^{\frac{1}{7} \cdot 7}}{{\sqrt[7]{3}}^{7}}$

Step 5.3.5.3

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

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{{(8 x)}^{\frac{7}{7}}}{{\sqrt[7]{3}}^{7}}$

Step 5.3.5.4

Cancel the common factor of $7$ .

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Step 5.3.5.4.1

Cancel the common factor.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{{(8 x)}^{\frac{7}{7}}}{{\sqrt[7]{3}}^{7}}$

Step 5.3.5.4.2

Rewrite the expression.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{8 x}{{\sqrt[7]{3}}^{7}}$

Step 5.3.5.5

Simplify.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{8 x}{{\sqrt[7]{3}}^{7}}$

Step 5.3.6

Rewrite ${\sqrt[7]{3}}^{7}$ as $3$ .

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Step 5.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{3}$ as $3^{\frac{1}{7}}$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{8 x}{{(3^{\frac{1}{7}})}^{7}}$

Step 5.3.6.2

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

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{8 x}{3^{\frac{1}{7} \cdot 7}}$

Step 5.3.6.3

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

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{8 x}{3^{\frac{7}{7}}}$

Step 5.3.6.4

Cancel the common factor of $7$ .

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Step 5.3.6.4.1

Cancel the common factor.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{8 x}{3^{\frac{7}{7}}}$

Step 5.3.6.4.2

Rewrite the expression.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{8 x}{3}$

Step 5.3.6.5

Evaluate the exponent.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = {(- 1)}^{8} (\frac{3}{8}) \cdot \frac{8 x}{3}$

Step 5.3.7

Simplify the expression.

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Step 5.3.7.1

Raise $- 1$ to the power of $8$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = 1 (\frac{3}{8}) \cdot \frac{8 x}{3}$

Step 5.3.7.2

Multiply $\frac{3}{8}$ by $1$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = \frac{3}{8} \cdot \frac{8 x}{3}$

Step 5.3.8

Cancel the common factor of $3$ .

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Step 5.3.8.1

Cancel the common factor.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = \frac{3}{8} \cdot \frac{8 x}{3}$

Step 5.3.8.2

Rewrite the expression.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = \frac{1}{8} \cdot (8 x)$

Step 5.3.9

Cancel the common factor of $8$ .

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Step 5.3.9.1

Factor $8$ out of $8 x$ .

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = \frac{1}{8} \cdot (8 (x))$

Step 5.3.9.2

Cancel the common factor.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = \frac{1}{8} \cdot (8 x)$

Step 5.3.9.3

Rewrite the expression.

$f (- \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}$ is the inverse of $f (x) = - \frac{3}{8} x^{7}$ .

$f^{- 1} (x) = - \frac{\sqrt[7]{8 x}}{\sqrt[7]{3}}$