Find the Inverse f(x)=(3x+2)^(1/2)

$f (x) = {(3 x + 2)}^{\frac{1}{2}}$

Step 1

Write $f (x) = {(3 x + 2)}^{\frac{1}{2}}$ as an equation.

$y = {(3 x + 2)}^{\frac{1}{2}}$

Step 2

Interchange the variables.

$x = {(3 y + 2)}^{\frac{1}{2}}$

Step 3

Solve for $y$ .

Rewrite the equation as ${(3 y + 2)}^{\frac{1}{2}} = x$ .

${(3 y + 2)}^{\frac{1}{2}} = x$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(3 y + 2)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(3 y + 2)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 y + 2)}^{\frac{1}{2}})}^{2}$ .

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

${(3 y + 2)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 y + 2)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(3 y + 2)}^{1} = x^{2}$

Simplify.

$3 y + 2 = x^{2}$

Solve for $y$ .

Subtract $2$ from both sides of the equation.

$3 y = x^{2} - 2$

Divide each term in $3 y = x^{2} - 2$ by $3$ and simplify.

Divide each term in $3 y = x^{2} - 2$ by $3$ .

$\frac{3 y}{3} = \frac{x^{2}}{3} + \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{x^{2}}{3} + \frac{- 2}{3}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{3} + \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{x^{2}}{3} - \frac{2}{3}$

Step 4

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

$f^{- 1} (x) = \frac{x^{2}}{3} - \frac{2}{3}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{2}{3}$ is the inverse of $f (x) = {(3 x + 2)}^{\frac{1}{2}}$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} ({(3 x + 2)}^{\frac{1}{2}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{{({(3 x + 2)}^{\frac{1}{2}})}^{2}}{3} - \frac{2}{3}$

Combine the numerators over the common denominator.

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{{({(3 x + 2)}^{\frac{1}{2}})}^{2} - 2}{3}$

Simplify each term.

Multiply the exponents in ${({(3 x + 2)}^{\frac{1}{2}})}^{2}$ .

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

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{{(3 x + 2)}^{\frac{1}{2} \cdot 2} - 2}{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{{(3 x + 2)}^{\frac{1}{2} \cdot 2} - 2}{3}$

Rewrite the expression.

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{(3 x + 2) - 2}{3}$

Simplify.

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{3 x + 2 - 2}{3}$

Simplify terms.

Combine the opposite terms in $3 x + 2 - 2$ .

Subtract $2$ from $2$ .

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{3 x + 0}{3}$

Add $3 x$ and $0$ .

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{3 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = \frac{3 x}{3}$

Divide $x$ by $1$ .

$f^{- 1} ({(3 x + 2)}^{\frac{1}{2}}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{2}}{3} - \frac{2}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(3 (\frac{x^{2}}{3} - \frac{2}{3}) + 2)}^{\frac{1}{2}}$

Simplify each term.

Apply the distributive property.

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(3 (\frac{x^{2}}{3}) + 3 (- \frac{2}{3}) + 2)}^{\frac{1}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(3 (\frac{x^{2}}{3}) + 3 (- \frac{2}{3}) + 2)}^{\frac{1}{2}}$

Rewrite the expression.

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(x^{2} + 3 (- \frac{2}{3}) + 2)}^{\frac{1}{2}}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(x^{2} + 3 (\frac{- 2}{3}) + 2)}^{\frac{1}{2}}$

Cancel the common factor.

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(x^{2} + 3 (\frac{- 2}{3}) + 2)}^{\frac{1}{2}}$

Rewrite the expression.

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(x^{2} - 2 + 2)}^{\frac{1}{2}}$

Simplify by adding terms.

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

Add $- 2$ and $2$ .

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(x^{2} + 0)}^{\frac{1}{2}}$

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

$f (\frac{x^{2}}{3} - \frac{2}{3}) = {(x^{2})}^{\frac{1}{2}}$

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

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

$f (\frac{x^{2}}{3} - \frac{2}{3}) = x^{2 (\frac{1}{2})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (\frac{x^{2}}{3} - \frac{2}{3}) = x^{2 (\frac{1}{2})}$

Rewrite the expression.

$f (\frac{x^{2}}{3} - \frac{2}{3}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{2}{3}$ is the inverse of $f (x) = {(3 x + 2)}^{\frac{1}{2}}$ .

$f^{- 1} (x) = \frac{x^{2}}{3} - \frac{2}{3}$

Find the Inverse f(x)=(3x+2)^(1/2)

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