Find the Inverse f(x) = square root of (x+3)/7

$f (x) = \sqrt{\frac{x + 3}{7}}$

Write $f (x) = \sqrt{\frac{x + 3}{7}}$ as an equation.

$y = \sqrt{\frac{x + 3}{7}}$

Interchange the variables.

$x = \sqrt{\frac{y + 3}{7}}$

Solve for $y$ .

Rewrite the equation as $\sqrt{\frac{y + 3}{7}} = x$ .

$\sqrt{\frac{y + 3}{7}} = x$

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

${\sqrt{\frac{y + 3}{7}}}^{2} = x^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{y + 3}{7}}$ as ${(\frac{y + 3}{7})}^{\frac{1}{2}}$ .

${({(\frac{y + 3}{7})}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(\frac{y + 3}{7})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(\frac{y + 3}{7})}^{\frac{1}{2}})}^{2}$ .

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

${(\frac{y + 3}{7})}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(\frac{y + 3}{7})}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(\frac{y + 3}{7})}^{1} = x^{2}$

Simplify.

$\frac{y + 3}{7} = x^{2}$

Solve for $y$ .

Multiply both sides of the equation by $7$ .

$7 \frac{y + 3}{7} = 7 x^{2}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$7 \frac{y + 3}{7} = 7 x^{2}$

Rewrite the expression.

$y + 3 = 7 x^{2}$

Subtract $3$ from both sides of the equation.

$y = 7 x^{2} - 3$

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

$f^{- 1} (x) = 7 x^{2} - 3$

Verify if $f^{- 1} (x) = 7 x^{2} - 3$ is the inverse of $f (x) = \sqrt{\frac{x + 3}{7}}$ .

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 $f^{- 1} (\sqrt{\frac{x + 3}{7}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\sqrt{\frac{x + 3}{7}})}^{2} - 3$

Simplify each term.

Rewrite $\sqrt{\frac{x + 3}{7}}$ as $\frac{\sqrt{x + 3}}{\sqrt{7}}$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3}}{\sqrt{7}})}^{2} - 3$

Multiply $\frac{\sqrt{x + 3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})}^{2} - 3$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x + 3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{\sqrt{7} \sqrt{7}})}^{2} - 3$

Raise $\sqrt{7}$ to the power of $1$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{\sqrt{7} \sqrt{7}})}^{2} - 3$

Raise $\sqrt{7}$ to the power of $1$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{\sqrt{7} \sqrt{7}})}^{2} - 3$

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

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}})}^{2} - 3$

Add $1$ and $1$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{{\sqrt{7}}^{2}})}^{2} - 3$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}})}^{2} - 3$

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

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}})}^{2} - 3$

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

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{7^{\frac{2}{2}}})}^{2} - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{7^{\frac{2}{2}}})}^{2} - 3$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{7})}^{2} - 3$

Evaluate the exponent.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{x + 3} \sqrt{7}}{7})}^{2} - 3$

Combine using the product rule for radicals.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 {(\frac{\sqrt{(x + 3) \cdot 7}}{7})}^{2} - 3$

Apply the product rule to $\frac{\sqrt{(x + 3) \cdot 7}}{7}$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{{\sqrt{(x + 3) \cdot 7}}^{2}}{7^{2}}) - 3$

Simplify the numerator.

Rewrite ${\sqrt{(x + 3) \cdot 7}}^{2}$ as $(x + 3) \cdot 7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(x + 3) \cdot 7}$ as ${((x + 3) \cdot 7)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{{({((x + 3) \cdot 7)}^{\frac{1}{2}})}^{2}}{7^{2}}) - 3$

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

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{{((x + 3) \cdot 7)}^{\frac{1}{2} \cdot 2}}{7^{2}}) - 3$

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

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{{((x + 3) \cdot 7)}^{\frac{2}{2}}}{7^{2}}) - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{{((x + 3) \cdot 7)}^{\frac{2}{2}}}{7^{2}}) - 3$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{(x + 3) \cdot 7}{7^{2}}) - 3$

Simplify.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{(x + 3) \cdot 7}{7^{2}}) - 3$

Apply the distributive property.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{x \cdot 7 + 3 \cdot 7}{7^{2}}) - 3$

Move $7$ to the left of $x$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{7 \cdot x + 3 \cdot 7}{7^{2}}) - 3$

Multiply $3$ by $7$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{7 \cdot x + 21}{7^{2}}) - 3$

Factor $7$ out of $7 x + 21$ .

Factor $7$ out of $7 x$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{7 (x) + 21}{7^{2}}) - 3$

Factor $7$ out of $21$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{7 x + 7 \cdot 3}{7^{2}}) - 3$

Factor $7$ out of $7 x + 7 \cdot 3$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{7 (x + 3)}{7^{2}}) - 3$

Raise $7$ to the power of $2$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{7 (x + 3)}{49}) - 3$

Cancel the common factor of $7$ .

Factor $7$ out of $49$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{7 (x + 3)}{7 (7)}) - 3$

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = 7 (\frac{7 (x + 3)}{7 \cdot 7}) - 3$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = \frac{7 (x + 3)}{7} - 3$

Cancel the common factor of $7$ .

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = \frac{7 (x + 3)}{7} - 3$

Divide $x + 3$ by $1$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = x + 3 - 3$

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

Subtract $3$ from $3$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\sqrt{\frac{x + 3}{7}}) = x$

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

Set up the composite result function.

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

Evaluate $f (7 x^{2} - 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (7 x^{2} - 3) = \sqrt{\frac{(7 x^{2} - 3) + 3}{7}}$

Add $- 3$ and $3$ .

$f (7 x^{2} - 3) = \sqrt{\frac{7 x^{2} + 0}{7}}$

Add $7 x^{2}$ and $0$ .

$f (7 x^{2} - 3) = \sqrt{\frac{7 x^{2}}{7}}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{7 x^{2}}{7}$ by cancelling the common factors.

Cancel the common factor.

$f (7 x^{2} - 3) = \sqrt{\frac{7 x^{2}}{7}}$

Rewrite the expression.

$f (7 x^{2} - 3) = \sqrt{\frac{x^{2}}{1}}$

Divide $x^{2}$ by $1$ .

$f (7 x^{2} - 3) = \sqrt{x^{2}}$

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

$f (7 x^{2} - 3) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 7 x^{2} - 3$ is the inverse of $f (x) = \sqrt{\frac{x + 3}{7}}$ .

$f^{- 1} (x) = 7 x^{2} - 3$

Find the Inverse f(x) = square root of (x+3)/7

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