Precalculus Examples

$f (x) = \sqrt{x + 4} - 2$

Step 1

Write $f (x) = \sqrt{x + 4} - 2$ as an equation.

$y = \sqrt{x + 4} - 2$

Step 2

Interchange the variables.

$x = \sqrt{y + 4} - 2$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\sqrt{y + 4} - 2 = x$ .

$\sqrt{y + 4} - 2 = x$

Step 3.2

Add $2$ to both sides of the equation.

$\sqrt{y + 4} = x + 2$

Step 3.3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y + 4}}^{2} = {(x + 2)}^{2}$

Step 3.4

Simplify each side of the equation.

Tap for more steps...

Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 4}$ as ${(y + 4)}^{\frac{1}{2}}$ .

${({(y + 4)}^{\frac{1}{2}})}^{2} = {(x + 2)}^{2}$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Simplify ${({(y + 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.4.2.1.1

Multiply the exponents in ${({(y + 4)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.4.2.1.1.1

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

${(y + 4)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Step 3.4.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.2.1.1.2.1

Cancel the common factor.

${(y + 4)}^{\frac{1}{2} \cdot 2} = {(x + 2)}^{2}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

${(y + 4)}^{1} = {(x + 2)}^{2}$

Step 3.4.2.1.2

Simplify.

$y + 4 = {(x + 2)}^{2}$

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Simplify ${(x + 2)}^{2}$ .

Tap for more steps...

Step 3.4.3.1.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$y + 4 = (x + 2) (x + 2)$

Step 3.4.3.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Tap for more steps...

Step 3.4.3.1.2.1

Apply the distributive property.

$y + 4 = x (x + 2) + 2 (x + 2)$

Step 3.4.3.1.2.2

Apply the distributive property.

$y + 4 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 3.4.3.1.2.3

Apply the distributive property.

$y + 4 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

Step 3.4.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.4.3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.4.3.1.3.1.1

Multiply $x$ by $x$ .

$y + 4 = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Step 3.4.3.1.3.1.2

Move $2$ to the left of $x$ .

$y + 4 = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Step 3.4.3.1.3.1.3

Multiply $2$ by $2$ .

$y + 4 = x^{2} + 2 x + 2 x + 4$

Step 3.4.3.1.3.2

Add $2 x$ and $2 x$ .

$y + 4 = x^{2} + 4 x + 4$

Step 3.5

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 3.5.1

Subtract $4$ from both sides of the equation.

$y = x^{2} + 4 x + 4 - 4$

Step 3.5.2

Combine the opposite terms in $x^{2} + 4 x + 4 - 4$ .

Tap for more steps...

Step 3.5.2.1

Subtract $4$ from $4$ .

$y = x^{2} + 4 x + 0$

Step 3.5.2.2

Add $x^{2} + 4 x$ and $0$ .

$y = x^{2} + 4 x$

Step 4

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

$f^{- 1} (x) = x^{2} + 4 x$

Step 5

Verify if $f^{- 1} (x) = x^{2} + 4 x$ is the inverse of $f (x) = \sqrt{x + 4} - 2$ .

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 $f^{- 1} (\sqrt{x + 4} - 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{x + 4} - 2) = {(\sqrt{x + 4} - 2)}^{2} + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Rewrite ${(\sqrt{x + 4} - 2)}^{2}$ as $(\sqrt{x + 4} - 2) (\sqrt{x + 4} - 2)$ .

$f^{- 1} (\sqrt{x + 4} - 2) = (\sqrt{x + 4} - 2) (\sqrt{x + 4} - 2) + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.2

Expand $(\sqrt{x + 4} - 2) (\sqrt{x + 4} - 2)$ using the FOIL Method.

Tap for more steps...

Step 5.2.3.2.1

Apply the distributive property.

$f^{- 1} (\sqrt{x + 4} - 2) = \sqrt{x + 4} (\sqrt{x + 4} - 2) - 2 (\sqrt{x + 4} - 2) + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.2.2

Apply the distributive property.

$f^{- 1} (\sqrt{x + 4} - 2) = \sqrt{x + 4} \sqrt{x + 4} + \sqrt{x + 4} \cdot - 2 - 2 (\sqrt{x + 4} - 2) + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.2.3

Apply the distributive property.

$f^{- 1} (\sqrt{x + 4} - 2) = \sqrt{x + 4} \sqrt{x + 4} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3

Simplify and combine like terms.

Tap for more steps...

Step 5.2.3.3.1

Simplify each term.

Tap for more steps...

Step 5.2.3.3.1.1

Multiply $\sqrt{x + 4} \sqrt{x + 4}$ .

Tap for more steps...

Step 5.2.3.3.1.1.1

Raise $\sqrt{x + 4}$ to the power of $1$ .

$f^{- 1} (\sqrt{x + 4} - 2) = \sqrt{x + 4} \sqrt{x + 4} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.1.2

Raise $\sqrt{x + 4}$ to the power of $1$ .

$f^{- 1} (\sqrt{x + 4} - 2) = \sqrt{x + 4} \sqrt{x + 4} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.1.3

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

$f^{- 1} (\sqrt{x + 4} - 2) = {\sqrt{x + 4}}^{1 + 1} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt{x + 4} - 2) = {\sqrt{x + 4}}^{2} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.2

Rewrite ${\sqrt{x + 4}}^{2}$ as $x + 4$ .

Tap for more steps...

Step 5.2.3.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 4}$ as ${(x + 4)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{x + 4} - 2) = {({(x + 4)}^{\frac{1}{2}})}^{2} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.2.2

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

$f^{- 1} (\sqrt{x + 4} - 2) = {(x + 4)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\sqrt{x + 4} - 2) = {(x + 4)}^{\frac{2}{2}} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.3.3.1.2.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{x + 4} - 2) = {(x + 4)}^{\frac{2}{2}} + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.2.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{x + 4} - 2) = (x + 4) + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.2.5

Simplify.

$f^{- 1} (\sqrt{x + 4} - 2) = x + 4 + \sqrt{x + 4} \cdot - 2 - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.3

Move $- 2$ to the left of $\sqrt{x + 4}$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x + 4 - 2 \cdot \sqrt{x + 4} - 2 \sqrt{x + 4} - 2 \cdot - 2 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.1.4

Multiply $- 2$ by $- 2$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x + 4 - 2 \sqrt{x + 4} - 2 \sqrt{x + 4} + 4 + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.2

Add $4$ and $4$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x + 8 - 2 \sqrt{x + 4} - 2 \sqrt{x + 4} + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.3.3

Subtract $2 \sqrt{x + 4}$ from $- 2 \sqrt{x + 4}$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x + 8 - 4 \sqrt{x + 4} + 4 (\sqrt{x + 4} - 2)$

Step 5.2.3.4

Apply the distributive property.

$f^{- 1} (\sqrt{x + 4} - 2) = x + 8 - 4 \sqrt{x + 4} + 4 \sqrt{x + 4} + 4 \cdot - 2$

Step 5.2.3.5

Multiply $4$ by $- 2$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x + 8 - 4 \sqrt{x + 4} + 4 \sqrt{x + 4} - 8$

Step 5.2.4

Combine the opposite terms in $x + 8 - 4 \sqrt{x + 4} + 4 \sqrt{x + 4} - 8$ .

Tap for more steps...

Step 5.2.4.1

Add $- 4 \sqrt{x + 4}$ and $4 \sqrt{x + 4}$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x + 8 + 0 - 8$

Step 5.2.4.2

Add $x + 8$ and $0$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x + 8 - 8$

Step 5.2.4.3

Subtract $8$ from $8$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x + 0$

Step 5.2.4.4

Add $x$ and $0$ .

$f^{- 1} (\sqrt{x + 4} - 2) = x$

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 $f (x^{2} + 4 x)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{2} + 4 x) = \sqrt{(x^{2} + 4 x) + 4} - 2$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

Factor using the perfect square rule.

Tap for more steps...

Step 5.3.3.1.1

Rewrite $4$ as $2^{2}$ .

$f (x^{2} + 4 x) = \sqrt{x^{2} + 4 x + 2^{2}} - 2$

Step 5.3.3.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Step 5.3.3.1.3

Rewrite the polynomial.

$f (x^{2} + 4 x) = \sqrt{x^{2} + 2 \cdot x \cdot 2 + 2^{2}} - 2$

Step 5.3.3.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2$ .

$f (x^{2} + 4 x) = \sqrt{{(x + 2)}^{2}} - 2$

Step 5.3.3.2

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

$f (x^{2} + 4 x) = x + 2 - 2$

Step 5.3.4

Combine the opposite terms in $x + 2 - 2$ .

Tap for more steps...

Step 5.3.4.1

Subtract $2$ from $2$ .

$f (x^{2} + 4 x) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (x^{2} + 4 x) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{2} + 4 x$ is the inverse of $f (x) = \sqrt{x + 4} - 2$ .

$f^{- 1} (x) = x^{2} + 4 x$