Trigonometry Examples

$F (x) = {(x - 6)}^{3}$

Step 1

Write $F (x) = {(x - 6)}^{3}$ as an equation.

$y = {(x - 6)}^{3}$

Step 2

Interchange the variables.

$x = {(y - 6)}^{3}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as ${(y - 6)}^{3} = x$ .

${(y - 6)}^{3} = x$

Step 3.2

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

$y - 6 = \sqrt[3]{x}$

Step 3.3

Add $6$ to both sides of the equation.

$y = \sqrt[3]{x} + 6$

Step 4

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

$F^{- 1} (x) = \sqrt[3]{x} + 6$

Step 5

Verify if $F^{- 1} (x) = \sqrt[3]{x} + 6$ is the inverse of $F (x) = {(x - 6)}^{3}$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $F^{- 1} ({(x - 6)}^{3})$ by substituting in the value of $F$ into $F^{- 1}$ .

$F^{- 1} ({(x - 6)}^{3}) = \sqrt[3]{{(x - 6)}^{3}} + 6$

Step 5.2.3

Remove parentheses.

$F^{- 1} ({(x - 6)}^{3}) = \sqrt[3]{{(x - 6)}^{3}} + 6$

Step 5.2.4

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

$F^{- 1} ({(x - 6)}^{3}) = x - 6 + 6$

Step 5.2.5

Combine the opposite terms in $x - 6 + 6$ .

Tap for more steps...

Step 5.2.5.1

Add $- 6$ and $6$ .

$F^{- 1} ({(x - 6)}^{3}) = x + 0$

Step 5.2.5.2

Add $x$ and $0$ .

$F^{- 1} ({(x - 6)}^{3}) = x$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $F (\sqrt[3]{x} + 6)$ by substituting in the value of $F^{- 1}$ into $F$ .

$F (\sqrt[3]{x} + 6) = {((\sqrt[3]{x} + 6) - 6)}^{3}$

Step 5.3.3

Combine the opposite terms in $(\sqrt[3]{x} + 6) - 6$ .

Tap for more steps...

Step 5.3.3.1

Subtract $6$ from $6$ .

$F (\sqrt[3]{x} + 6) = {(\sqrt[3]{x} + 0)}^{3}$

Step 5.3.3.2

Add $\sqrt[3]{x}$ and $0$ .

$F (\sqrt[3]{x} + 6) = {\sqrt[3]{x}}^{3}$

Step 5.3.4

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

Tap for more steps...

Step 5.3.4.1

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

$F (\sqrt[3]{x} + 6) = {(x^{\frac{1}{3}})}^{3}$

Step 5.3.4.2

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

$F (\sqrt[3]{x} + 6) = x^{\frac{1}{3} \cdot 3}$

Step 5.3.4.3

Combine $\frac{1}{3}$ and $3$ .

$F (\sqrt[3]{x} + 6) = x^{\frac{3}{3}}$

Step 5.3.4.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.4.4.1

Cancel the common factor.

$F (\sqrt[3]{x} + 6) = x^{\frac{3}{3}}$

Step 5.3.4.4.2

Rewrite the expression.

$F (\sqrt[3]{x} + 6) = x$

Step 5.3.4.5

Simplify.

$F (\sqrt[3]{x} + 6) = x$

Step 5.4

Since $F^{- 1} (F (x)) = x$ and $F (F^{- 1} (x)) = x$ , then $F^{- 1} (x) = \sqrt[3]{x} + 6$ is the inverse of $F (x) = {(x - 6)}^{3}$ .

$F^{- 1} (x) = \sqrt[3]{x} + 6$