Find the Inverse square root of 8x+10

$\sqrt{8 x + 10}$

Step 1

Interchange the variables.

$x = \sqrt{8 y + 10}$

Step 2

Solve for $y$ .

Rewrite the equation as $\sqrt{8 y + 10} = x$ .

$\sqrt{8 y + 10} = x$

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

${\sqrt{8 y + 10}}^{2} = x^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 y + 10}$ as ${(8 y + 10)}^{\frac{1}{2}}$ .

${({(8 y + 10)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(8 y + 10)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(8 y + 10)}^{\frac{1}{2}})}^{2}$ .

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

${(8 y + 10)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(8 y + 10)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(8 y + 10)}^{1} = x^{2}$

Simplify.

$8 y + 10 = x^{2}$

Solve for $y$ .

Subtract $10$ from both sides of the equation.

$8 y = x^{2} - 10$

Divide each term in $8 y = x^{2} - 10$ by $8$ and simplify.

Divide each term in $8 y = x^{2} - 10$ by $8$ .

$\frac{8 y}{8} = \frac{x^{2}}{8} + \frac{- 10}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 y}{8} = \frac{x^{2}}{8} + \frac{- 10}{8}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{8} + \frac{- 10}{8}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 10$ and $8$ .

Factor $2$ out of $- 10$ .

$y = \frac{x^{2}}{8} + \frac{2 (- 5)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$y = \frac{x^{2}}{8} + \frac{2 \cdot - 5}{2 \cdot 4}$

Cancel the common factor.

$y = \frac{x^{2}}{8} + \frac{2 \cdot - 5}{2 \cdot 4}$

Rewrite the expression.

$y = \frac{x^{2}}{8} + \frac{- 5}{4}$

Move the negative in front of the fraction.

$y = \frac{x^{2}}{8} - \frac{5}{4}$

Step 3

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

$f^{- 1} (x) = \frac{x^{2}}{8} - \frac{5}{4}$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{2}}{8} - \frac{5}{4}$ is the inverse of $f (x) = \sqrt{8 x + 10}$ .

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 $f^{- 1} (\sqrt{8 x + 10})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{{(\sqrt{8 x + 10})}^{2}}{8} - \frac{5}{4}$

Simplify each term.

Simplify the numerator.

Factor $2$ out of $8 x + 10$ .

Factor $2$ out of $8 x$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{{\sqrt{2 (4 x) + 10}}^{2}}{8} - \frac{5}{4}$

Factor $2$ out of $10$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{{\sqrt{2 (4 x) + 2 (5)}}^{2}}{8} - \frac{5}{4}$

Factor $2$ out of $2 (4 x) + 2 (5)$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{{\sqrt{2 (4 x + 5)}}^{2}}{8} - \frac{5}{4}$

Rewrite ${\sqrt{2 (4 x + 5)}}^{2}$ as $2 (4 x + 5)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (4 x + 5)}$ as ${(2 (4 x + 5))}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{{({(2 (4 x + 5))}^{\frac{1}{2}})}^{2}}{8} - \frac{5}{4}$

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

$f^{- 1} (\sqrt{8 x + 10}) = \frac{{(2 (4 x + 5))}^{\frac{1}{2} \cdot 2}}{8} - \frac{5}{4}$

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

$f^{- 1} (\sqrt{8 x + 10}) = \frac{{(2 (4 x + 5))}^{\frac{2}{2}}}{8} - \frac{5}{4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{8 x + 10}) = \frac{{(2 (4 x + 5))}^{\frac{2}{2}}}{8} - \frac{5}{4}$

Rewrite the expression.

$f^{- 1} (\sqrt{8 x + 10}) = \frac{2 (4 x + 5)}{8} - \frac{5}{4}$

Simplify.

$f^{- 1} (\sqrt{8 x + 10}) = \frac{2 (4 x + 5)}{8} - \frac{5}{4}$

Apply the distributive property.

$f^{- 1} (\sqrt{8 x + 10}) = \frac{2 (4 x) + 2 \cdot 5}{8} - \frac{5}{4}$

Multiply $4$ by $2$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{8 x + 2 \cdot 5}{8} - \frac{5}{4}$

Multiply $2$ by $5$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{8 x + 10}{8} - \frac{5}{4}$

Factor $2$ out of $8 x + 10$ .

Factor $2$ out of $8 x$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{2 (4 x) + 10}{8} - \frac{5}{4}$

Factor $2$ out of $10$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{2 (4 x) + 2 (5)}{8} - \frac{5}{4}$

Factor $2$ out of $2 (4 x) + 2 (5)$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{2 (4 x + 5)}{8} - \frac{5}{4}$

Cancel the common factors.

Factor $2$ out of $8$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{2 (4 x + 5)}{2 \cdot 4} - \frac{5}{4}$

Cancel the common factor.

$f^{- 1} (\sqrt{8 x + 10}) = \frac{2 (4 x + 5)}{2 \cdot 4} - \frac{5}{4}$

Rewrite the expression.

$f^{- 1} (\sqrt{8 x + 10}) = \frac{4 x + 5}{4} - \frac{5}{4}$

Simplify terms.

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt{8 x + 10}) = \frac{4 x + 5 - 5}{4}$

Combine the opposite terms in $4 x + 5 - 5$ .

Subtract $5$ from $5$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{4 x + 0}{4}$

Add $4 x$ and $0$ .

$f^{- 1} (\sqrt{8 x + 10}) = \frac{4 x}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (\sqrt{8 x + 10}) = \frac{4 x}{4}$

Divide $x$ by $1$ .

$f^{- 1} (\sqrt{8 x + 10}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{2}}{8} - \frac{5}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{8 (\frac{x^{2}}{8} - \frac{5}{4}) + 10}$

Factor $2$ out of $8 (\frac{x^{2}}{8} - \frac{5}{4}) + 10$ .

Factor $2$ out of $8 (\frac{x^{2}}{8} - \frac{5}{4})$ .

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (4 (\frac{x^{2}}{8} - \frac{5}{4})) + 10}$

Factor $2$ out of $10$ .

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (4 (\frac{x^{2}}{8} - \frac{5}{4})) + 2 (5)}$

Factor $2$ out of $2 (4 (\frac{x^{2}}{8} - \frac{5}{4})) + 2 (5)$ .

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (4 (\frac{x^{2}}{8} - \frac{5}{4}) + 5)}$

Apply the distributive property.

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (4 (\frac{x^{2}}{8}) + 4 (- \frac{5}{4}) + 5)}$

Cancel the common factor of $4$ .

Factor $4$ out of $8$ .

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (4 (\frac{x^{2}}{4 (2)}) + 4 (- \frac{5}{4}) + 5)}$

Cancel the common factor.

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (4 (\frac{x^{2}}{4 \cdot 2}) + 4 (- \frac{5}{4}) + 5)}$

Rewrite the expression.

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (\frac{x^{2}}{2} + 4 (- \frac{5}{4}) + 5)}$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (\frac{x^{2}}{2} + 4 (\frac{- 5}{4}) + 5)}$

Cancel the common factor.

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (\frac{x^{2}}{2} + 4 (\frac{- 5}{4}) + 5)}$

Rewrite the expression.

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (\frac{x^{2}}{2} - 5 + 5)}$

Simplify by adding numbers.

Add $- 5$ and $5$ .

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (\frac{x^{2}}{2} + 0)}$

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

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{2 (\frac{x^{2}}{2})}$

Combine $2$ and $\frac{x^{2}}{2}$ .

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{\frac{2 x^{2}}{2}}$

Reduce the expression by cancelling the common factors.

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

Cancel the common factor.

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{\frac{2 x^{2}}{2}}$

Rewrite the expression.

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{\frac{x^{2}}{1}}$

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

$f (\frac{x^{2}}{8} - \frac{5}{4}) = \sqrt{x^{2}}$

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

$f (\frac{x^{2}}{8} - \frac{5}{4}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{8} - \frac{5}{4}$ is the inverse of $f (x) = \sqrt{8 x + 10}$ .

$f^{- 1} (x) = \frac{x^{2}}{8} - \frac{5}{4}$

Find the Inverse square root of 8x+10

Related Questions

Support LCMGCF.com by disabling your adblocker.