Precalculus Examples

$f (x) = {(x + 4)}^{\frac{1}{4}}$ ; find $f^{- 1} (x)$

Step 1

Write $f (x) = {(x + 4)}^{\frac{1}{4}}$ as an equation.

$y = {(x + 4)}^{\frac{1}{4}}$

Step 2

Interchange the variables.

$x = {(y + 4)}^{\frac{1}{4}}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as ${(y + 4)}^{\frac{1}{4}} = x$ .

${(y + 4)}^{\frac{1}{4}} = x$

Step 3.2

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${({(y + 4)}^{\frac{1}{4}})}^{4} = x^{4}$

Step 3.3

Simplify the left side.

Tap for more steps...

Step 3.3.1

Simplify ${({(y + 4)}^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 3.3.1.1

Multiply the exponents in ${({(y + 4)}^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 3.3.1.1.1

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

${(y + 4)}^{\frac{1}{4} \cdot 4} = x^{4}$

Step 3.3.1.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.3.1.1.2.1

Cancel the common factor.

${(y + 4)}^{\frac{1}{4} \cdot 4} = x^{4}$

Step 3.3.1.1.2.2

Rewrite the expression.

${(y + 4)}^{1} = x^{4}$

Step 3.3.1.2

Simplify.

$y + 4 = x^{4}$

Step 3.4

Subtract $4$ from both sides of the equation.

$y = x^{4} - 4$

Step 4

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

$f^{- 1} (x) = x^{4} - 4$

Step 5

Verify if $f^{- 1} (x) = x^{4} - 4$ is the inverse of $f (x) = {(x + 4)}^{\frac{1}{4}}$ .

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 $f^{- 1} ({(x + 4)}^{\frac{1}{4}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(x + 4)}^{\frac{1}{4}}) = {({(x + 4)}^{\frac{1}{4}})}^{4} - 4$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Multiply the exponents in ${({(x + 4)}^{\frac{1}{4}})}^{4}$ .

Tap for more steps...

Step 5.2.3.1.1

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

$f^{- 1} ({(x + 4)}^{\frac{1}{4}}) = {(x + 4)}^{\frac{1}{4} \cdot 4} - 4$

Step 5.2.3.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.2.3.1.2.1

Cancel the common factor.

$f^{- 1} ({(x + 4)}^{\frac{1}{4}}) = {(x + 4)}^{\frac{1}{4} \cdot 4} - 4$

Step 5.2.3.1.2.2

Rewrite the expression.

$f^{- 1} ({(x + 4)}^{\frac{1}{4}}) = (x + 4) - 4$

Step 5.2.3.2

Simplify.

$f^{- 1} ({(x + 4)}^{\frac{1}{4}}) = x + 4 - 4$

Step 5.2.4

Combine the opposite terms in $x + 4 - 4$ .

Tap for more steps...

Step 5.2.4.1

Subtract $4$ from $4$ .

$f^{- 1} ({(x + 4)}^{\frac{1}{4}}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} ({(x + 4)}^{\frac{1}{4}}) = x$

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 $f (x^{4} - 4)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{4} - 4) = {((x^{4} - 4) + 4)}^{\frac{1}{4}}$

Step 5.3.3

Combine the opposite terms in $(x^{4} - 4) + 4$ .

Tap for more steps...

Step 5.3.3.1

Add $- 4$ and $4$ .

$f (x^{4} - 4) = {(x^{4} + 0)}^{\frac{1}{4}}$

Step 5.3.3.2

Add $x^{4}$ and $0$ .

$f (x^{4} - 4) = {(x^{4})}^{\frac{1}{4}}$

Step 5.3.4

Multiply the exponents in ${(x^{4})}^{\frac{1}{4}}$ .

Tap for more steps...

Step 5.3.4.1

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

$f (x^{4} - 4) = x^{4 (\frac{1}{4})}$

Step 5.3.4.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.3.4.2.1

Cancel the common factor.

$f (x^{4} - 4) = x^{4 (\frac{1}{4})}$

Step 5.3.4.2.2

Rewrite the expression.

$f (x^{4} - 4) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{4} - 4$ is the inverse of $f (x) = {(x + 4)}^{\frac{1}{4}}$ .

$f^{- 1} (x) = x^{4} - 4$