Calculus Examples

$f (x) = \sqrt{8 - 4 x}$

Step 1

Write $f (x) = \sqrt{8 - 4 x}$ as an equation.

$y = \sqrt{8 - 4 x}$

Step 2

Interchange the variables.

$x = \sqrt{8 - 4 y}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\sqrt{8 - 4 y} = x$ .

$\sqrt{8 - 4 y} = x$

Step 3.2

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

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

Step 3.3

Simplify each side of the equation.

Tap for more steps...

Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 - 4 y}$ as ${(8 - 4 y)}^{\frac{1}{2}}$ .

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

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Simplify ${({(8 - 4 y)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.3.2.1.1

Multiply the exponents in ${({(8 - 4 y)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 3.3.2.1.1.1

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

${(8 - 4 y)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1.1.2.1

Cancel the common factor.

${(8 - 4 y)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(8 - 4 y)}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$8 - 4 y = x^{2}$

Step 3.4

Solve for $y$ .

Tap for more steps...

Step 3.4.1

Subtract $8$ from both sides of the equation.

$- 4 y = x^{2} - 8$

Step 3.4.2

Divide each term in $- 4 y = x^{2} - 8$ by $- 4$ and simplify.

Tap for more steps...

Step 3.4.2.1

Divide each term in $- 4 y = x^{2} - 8$ by $- 4$ .

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

Step 3.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.4.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

Tap for more steps...

Step 3.4.2.3.1

Simplify each term.

Tap for more steps...

Step 3.4.2.3.1.1

Move the negative in front of the fraction.

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

Step 3.4.2.3.1.2

Divide $- 8$ by $- 4$ .

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

Step 4

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

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

Step 5

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

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1.1

Factor $4$ out of $8 - 4 x$ .

Tap for more steps...

Step 5.2.3.1.1.1

Factor $4$ out of $8$ .

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

Step 5.2.3.1.1.2

Factor $4$ out of $- 4 x$ .

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

Step 5.2.3.1.1.3

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

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

Step 5.2.3.1.2

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

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

Step 5.2.3.1.3

Pull terms out from under the radical.

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

Step 5.2.3.1.4

Apply the product rule to $2 \sqrt{2 - x}$ .

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

Step 5.2.3.1.5

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

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

Step 5.2.3.1.6

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

Tap for more steps...

Step 5.2.3.1.6.1

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

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

Step 5.2.3.1.6.2

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

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

Step 5.2.3.1.6.3

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

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

Step 5.2.3.1.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.3.1.6.4.1

Cancel the common factor.

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

Step 5.2.3.1.6.4.2

Rewrite the expression.

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

Step 5.2.3.1.6.5

Simplify.

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

Step 5.2.3.1.7

Apply the distributive property.

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

Step 5.2.3.1.8

Multiply $4$ by $2$ .

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

Step 5.2.3.1.9

Multiply $- 1$ by $4$ .

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

Step 5.2.3.1.10

Factor $4$ out of $8 - 4 x$ .

Tap for more steps...

Step 5.2.3.1.10.1

Factor $4$ out of $8$ .

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

Step 5.2.3.1.10.2

Factor $4$ out of $- 4 x$ .

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

Step 5.2.3.1.10.3

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

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

Step 5.2.3.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 5.2.3.2.1

Cancel the common factor.

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

Step 5.2.3.2.2

Divide $2 - x$ by $1$ .

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

Step 5.2.3.3

Apply the distributive property.

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

Step 5.2.3.4

Multiply $- 1$ by $2$ .

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

Step 5.2.3.5

Multiply $- - x$ .

Tap for more steps...

Step 5.2.3.5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.3.5.2

Multiply $x$ by $1$ .

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

Step 5.2.4

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

Tap for more steps...

Step 5.2.4.1

Add $- 2$ and $2$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

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

Tap for more steps...

Step 5.3.3.1

Factor $4$ out of $8$ .

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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

Step 5.3.4

Apply the distributive property.

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

Step 5.3.5

Simplify the expression.

Tap for more steps...

Step 5.3.5.1

Multiply $- 1$ by $2$ .

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

Step 5.3.5.2

Subtract $2$ from $2$ .

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

Step 5.3.5.3

Add $\frac{x^{2}}{4}$ and $0$ .

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

Step 5.3.6

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

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

Step 5.3.7

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.3.7.1

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

Tap for more steps...

Step 5.3.7.1.1

Cancel the common factor.

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

Step 5.3.7.1.2

Rewrite the expression.

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

Step 5.3.7.2

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

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

Step 5.3.8

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

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

Step 5.4

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

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