Trigonometry Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as ${(y - 1)}^{3} + 1 = x$ .

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

Step 2.2

Subtract $1$ from both sides of the equation.

${(y - 1)}^{3} = x - 1$

Step 2.3

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

$y - 1 = \sqrt[3]{x - 1}$

Step 2.4

Add $1$ to both sides of the equation.

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

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Combine the opposite terms in $\sqrt[3]{({(x - 1)}^{3} + 1) - 1} + 1$ .

Tap for more steps...

Step 4.2.3.1

Subtract $1$ from $1$ .

$f^{- 1} ({(x - 1)}^{3} + 1) = \sqrt[3]{{(x - 1)}^{3} + 0} + 1$

Step 4.2.3.2

Add ${(x - 1)}^{3}$ and $0$ .

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

Step 4.2.4

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

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

Step 4.2.5

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

Tap for more steps...

Step 4.2.5.1

Add $- 1$ and $1$ .

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

Step 4.2.5.2

Add $x$ and $0$ .

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

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Combine the opposite terms in ${((\sqrt[3]{x - 1} + 1) - 1)}^{3} + 1$ .

Tap for more steps...

Step 4.3.3.1

Subtract $1$ from $1$ .

$f (\sqrt[3]{x - 1} + 1) = {(\sqrt[3]{x - 1} + 0)}^{3} + 1$

Step 4.3.3.2

Add $\sqrt[3]{x - 1}$ and $0$ .

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

Step 4.3.4

Rewrite ${\sqrt[3]{x - 1}}^{3}$ as $x - 1$ .

Tap for more steps...

Step 4.3.4.1

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

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

Step 4.3.4.2

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

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

Step 4.3.4.3

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

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

Step 4.3.4.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.3.4.4.1

Cancel the common factor.

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

Step 4.3.4.4.2

Rewrite the expression.

$f (\sqrt[3]{x - 1} + 1) = (x - 1) + 1$

Step 4.3.4.5

Simplify.

$f (\sqrt[3]{x - 1} + 1) = x - 1 + 1$

Step 4.3.5

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

Tap for more steps...

Step 4.3.5.1

Add $- 1$ and $1$ .

$f (\sqrt[3]{x - 1} + 1) = x + 0$

Step 4.3.5.2

Add $x$ and $0$ .

$f (\sqrt[3]{x - 1} + 1) = x$

Step 4.4

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

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