Calculus Examples

$- \sqrt{x}$

Interchange the variables.

$x = - \sqrt{y}$

Solve for $y$ .

Tap for more steps...

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

$- \sqrt{y} = x$

Divide each term in $- \sqrt{y} = x$ by $- 1$ and simplify.

Tap for more steps...

Divide each term in $- \sqrt{y} = x$ by $- 1$ .

$\frac{- \sqrt{y}}{- 1} = \frac{x}{- 1}$

Simplify the left side.

Tap for more steps...

Dividing two negative values results in a positive value.

$\frac{\sqrt{y}}{1} = \frac{x}{- 1}$

Divide $\sqrt{y}$ by $1$ .

$\sqrt{y} = \frac{x}{- 1}$

Simplify the right side.

Tap for more steps...

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$\sqrt{y} = - 1 \cdot x$

Rewrite $- 1 \cdot x$ as $- x$ .

$\sqrt{y} = - x$

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

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

Simplify each side of the equation.

Tap for more steps...

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

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

Simplify the left side.

Tap for more steps...

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

Tap for more steps...

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

Tap for more steps...

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

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

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

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

Rewrite the expression.

$y^{1} = {(- x)}^{2}$

Simplify.

$y = {(- x)}^{2}$

Simplify the right side.

Tap for more steps...

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

Tap for more steps...

Apply the product rule to $- x$ .

$y = {(- 1)}^{2} x^{2}$

Raise $- 1$ to the power of $2$ .

$y = 1 x^{2}$

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

$y = x^{2}$

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

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

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

Tap for more steps...

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

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

Tap for more steps...

Set up the composite result function.

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

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

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

Simplify the expression.

Tap for more steps...

Apply the product rule to $- \sqrt{x}$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply ${\sqrt{x}}^{2}$ by $1$ .

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

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

Tap for more steps...

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

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

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

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

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

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

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

Tap for more steps...

Set up the composite result function.

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

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

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

Remove parentheses.

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

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

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

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

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