Finite Math Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Add $2$ to both sides of the equation.

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

Step 2.3

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

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

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

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

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

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

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

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Step 2.4.2.1.1

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

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Step 2.4.2.1.1.1

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

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

Step 2.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

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

Step 2.4.2.1.1.2.2

Rewrite the expression.

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

Step 2.4.2.1.2

Simplify.

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

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

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

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Step 2.4.3.1.1

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

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

Step 2.4.3.1.2

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

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Step 2.4.3.1.2.1

Apply the distributive property.

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

Step 2.4.3.1.2.2

Apply the distributive property.

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

Step 2.4.3.1.2.3

Apply the distributive property.

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

Step 2.4.3.1.3

Simplify and combine like terms.

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Step 2.4.3.1.3.1

Simplify each term.

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Step 2.4.3.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.4.3.1.3.1.2

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

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

Step 2.4.3.1.3.1.3

Multiply $2$ by $2$ .

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

Step 2.4.3.1.3.2

Add $2 x$ and $2 x$ .

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

Step 2.5

Divide each term in $4 y = x^{2} + 4 x + 4$ by $4$ and simplify.

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Step 2.5.1

Divide each term in $4 y = x^{2} + 4 x + 4$ by $4$ .

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

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $4$ .

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Step 2.5.2.1.1

Cancel the common factor.

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

Step 2.5.2.1.2

Divide $y$ by $1$ .

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

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Cancel the common factor of $4$ .

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Step 2.5.3.1.1.1

Cancel the common factor.

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

Step 2.5.3.1.1.2

Divide $x$ by $1$ .

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

Step 2.5.3.1.2

Divide $4$ by $4$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Remove parentheses.

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

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

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

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

Step 4.2.4.1.2

Apply the product rule to $4 x$ .

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

Step 4.2.4.1.3

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

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

Step 4.2.4.1.4

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

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

Step 4.2.4.1.5

Cancel the common factor of $2$ .

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Step 4.2.4.1.5.1

Cancel the common factor.

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

Step 4.2.4.1.5.2

Rewrite the expression.

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

Step 4.2.4.1.6

Evaluate the exponent.

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

Step 4.2.4.1.7

Factor $2$ out of $2 x^{\frac{1}{2}} - 2$ .

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Step 4.2.4.1.7.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 4.2.4.1.7.2

Factor $2$ out of $- 2$ .

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

Step 4.2.4.1.7.3

Factor $2$ out of $2 (x^{\frac{1}{2}}) + 2 (- 1)$ .

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

Step 4.2.4.1.8

Apply the product rule to $2 (x^{\frac{1}{2}} - 1)$ .

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

Step 4.2.4.1.9

Raise $2$ to the power of $2$ .

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

Step 4.2.4.2

Cancel the common factor.

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

Step 4.2.4.3

Divide ${(x^{\frac{1}{2}} - 1)}^{2}$ by $1$ .

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

Step 4.2.4.4

Rewrite ${(x^{\frac{1}{2}} - 1)}^{2}$ as $(x^{\frac{1}{2}} - 1) (x^{\frac{1}{2}} - 1)$ .

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

Step 4.2.4.5

Expand $(x^{\frac{1}{2}} - 1) (x^{\frac{1}{2}} - 1)$ using the FOIL Method.

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Step 4.2.4.5.1

Apply the distributive property.

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

Step 4.2.4.5.2

Apply the distributive property.

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

Step 4.2.4.5.3

Apply the distributive property.

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

Step 4.2.4.6

Simplify and combine like terms.

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Step 4.2.4.6.1

Simplify each term.

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Step 4.2.4.6.1.1

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.4.6.1.1.1

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

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

Step 4.2.4.6.1.1.2

Combine the numerators over the common denominator.

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

Step 4.2.4.6.1.1.3

Add $1$ and $1$ .

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

Step 4.2.4.6.1.1.4

Divide $2$ by $2$ .

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

Step 4.2.4.6.1.2

Simplify $x^{1}$ .

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

Step 4.2.4.6.1.3

Move $- 1$ to the left of $x^{\frac{1}{2}}$ .

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

Step 4.2.4.6.1.4

Rewrite $- 1 x^{\frac{1}{2}}$ as $- x^{\frac{1}{2}}$ .

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

Step 4.2.4.6.1.5

Rewrite $- 1 x^{\frac{1}{2}}$ as $- x^{\frac{1}{2}}$ .

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

Step 4.2.4.6.1.6

Multiply $- 1$ by $- 1$ .

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

Step 4.2.4.6.2

Subtract $x^{\frac{1}{2}}$ from $- x^{\frac{1}{2}}$ .

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

Step 4.2.4.7

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

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

Step 4.2.4.8

Pull terms out from under the radical.

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

Step 4.2.5

Simplify by adding terms.

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Step 4.2.5.1

Subtract $2$ from $1$ .

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

Step 4.2.5.2

Combine the opposite terms in $x - 2 x^{\frac{1}{2}} - 1 + 2 \sqrt{x} + 1$ .

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Step 4.2.5.2.1

Add $- 1$ and $1$ .

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

Step 4.2.5.2.2

Add $x - 2 x^{\frac{1}{2}}$ and $0$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Factor using the perfect square rule.

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Step 4.3.3.1.1

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

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

Step 4.3.3.1.2

Rewrite $\frac{x^{2}}{2^{2}}$ as ${(\frac{x}{2})}^{2}$ .

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

Step 4.3.3.1.3

Rewrite $1$ as $1^{2}$ .

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

Step 4.3.3.1.4

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

$x = 2 \cdot \frac{x}{2} \cdot 1$

Step 4.3.3.1.5

Rewrite the polynomial.

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

Step 4.3.3.1.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \frac{x}{2}$ and $b = 1$ .

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

Step 4.3.3.2

Write $1$ as a fraction with a common denominator.

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

Step 4.3.3.3

Combine the numerators over the common denominator.

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

Step 4.3.3.4

Apply the product rule to $\frac{x + 2}{2}$ .

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

Step 4.3.3.5

Raise $2$ to the power of $2$ .

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

Step 4.3.3.6

Combine $4$ and $\frac{{(x + 2)}^{2}}{4}$ .

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

Step 4.3.3.7

Reduce the expression by cancelling the common factors.

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Step 4.3.3.7.1

Reduce the expression $\frac{4 {(x + 2)}^{2}}{4}$ by cancelling the common factors.

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Step 4.3.3.7.1.1

Cancel the common factor.

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

Step 4.3.3.7.1.2

Rewrite the expression.

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

Step 4.3.3.7.2

Divide ${(x + 2)}^{2}$ by $1$ .

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

Step 4.3.3.8

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

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

Step 4.3.4

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

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Step 4.3.4.1

Subtract $2$ from $2$ .

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

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

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

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