Precalculus Examples

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

Step 1

Rewrite the equation as $y^{2} - 4 y = x$ .

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

Step 2

Subtract $x$ from both sides of the equation.

$y^{2} - 4 y - x = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = - 4$ , and $c = - x$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot (- x))}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot - 1 x$ .

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Step 5.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- 1 x)}}{2 \cdot 1}$

Step 5.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 1 x$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- 1 x))}}{2 \cdot 1}$

Step 5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot - 1 x)$ .

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

Step 5.1.2

Multiply $- 1$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - x)}}{2 \cdot 1}$

Step 5.1.3

Rewrite $- 4 (- 4 - x)$ as $2^{2} \cdot (- 1 (- 4 - x))$ .

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Step 5.1.3.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (- 4 - x)}}{2 \cdot 1}$

Step 5.1.3.2

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

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

Step 5.1.3.3

Add parentheses.

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

Step 5.1.4

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$

Step 5.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (- 4 - x)}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot - 1 x$ .

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Step 6.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- 1 x)}}{2 \cdot 1}$

Step 6.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 1 x$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- 1 x))}}{2 \cdot 1}$

Step 6.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot - 1 x)$ .

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

Step 6.1.2

Multiply $- 1$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - x)}}{2 \cdot 1}$

Step 6.1.3

Rewrite $- 4 (- 4 - x)$ as $2^{2} \cdot (- 1 (- 4 - x))$ .

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Step 6.1.3.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (- 4 - x)}}{2 \cdot 1}$

Step 6.1.3.2

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

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

Step 6.1.3.3

Add parentheses.

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

Step 6.1.4

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$

Step 6.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (- 4 - x)}$

Step 6.4

Change the $\pm$ to $+$ .

$y = 2 + \sqrt{- 1 (- 4 - x)}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot - 1 x$ .

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Step 7.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- 1 x)}}{2 \cdot 1}$

Step 7.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot - 1 x$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- 1 x))}}{2 \cdot 1}$

Step 7.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot - 1 x)$ .

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

Step 7.1.2

Multiply $- 1$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - x)}}{2 \cdot 1}$

Step 7.1.3

Rewrite $- 4 (- 4 - x)$ as $2^{2} \cdot (- 1 (- 4 - x))$ .

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Step 7.1.3.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (- 4 - x)}}{2 \cdot 1}$

Step 7.1.3.2

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

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

Step 7.1.3.3

Add parentheses.

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

Step 7.1.4

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$

Step 7.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 - x)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (- 4 - x)}$

Step 7.4

Change the $\pm$ to $-$ .

$y = 2 - \sqrt{- 1 (- 4 - x)}$

Step 8

The final answer is the combination of both solutions.

$y = 2 + \sqrt{- 1 (- 4 - x)}$

$y = 2 - \sqrt{- 1 (- 4 - x)}$

Step 9

Interchange the variables. Create an equation for each expression.

$x = 2 + \sqrt{- 1 (- 4 - y)}, x = 2 - \sqrt{- 1 (- 4 - y)}$

Step 10

Solve for $y$ .

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Step 10.1

Rewrite the equation as $2 + \sqrt{- 1 (- 4 - y)} = x$ .

$2 + \sqrt{- 1 (- 4 - y)} = x$

Step 10.2

Subtract $2$ from both sides of the equation.

$\sqrt{- 1 (- 4 - y)} = x - 2$

Step 10.3

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

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

Step 10.4

Simplify each side of the equation.

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Step 10.4.1

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

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

Step 10.4.2

Simplify the left side.

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Step 10.4.2.1

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

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Step 10.4.2.1.1

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

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Step 10.4.2.1.1.1

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

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

Step 10.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 10.4.2.1.1.2.1

Cancel the common factor.

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

Step 10.4.2.1.1.2.2

Rewrite the expression.

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

Step 10.4.2.1.2

Apply the distributive property.

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

Step 10.4.2.1.3

Multiply $- 1$ by $- 4$ .

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

Step 10.4.2.1.4

Multiply $- 1 (- y)$ .

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Step 10.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 10.4.2.1.4.2

Multiply $y$ by $1$ .

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

Step 10.4.2.1.5

Simplify.

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

Step 10.4.3

Simplify the right side.

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Step 10.4.3.1

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

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Step 10.4.3.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$4 + y = (x - 2) (x - 2)$

Step 10.4.3.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 10.4.3.1.2.1

Apply the distributive property.

$4 + y = x (x - 2) - 2 (x - 2)$

Step 10.4.3.1.2.2

Apply the distributive property.

$4 + y = x \cdot x + x \cdot - 2 - 2 (x - 2)$

Step 10.4.3.1.2.3

Apply the distributive property.

$4 + y = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 10.4.3.1.3

Simplify and combine like terms.

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Step 10.4.3.1.3.1

Simplify each term.

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Step 10.4.3.1.3.1.1

Multiply $x$ by $x$ .

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

Step 10.4.3.1.3.1.2

Move $- 2$ to the left of $x$ .

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

Step 10.4.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 10.4.3.1.3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 10.5

Move all terms not containing $y$ to the right side of the equation.

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Step 10.5.1

Subtract $4$ from both sides of the equation.

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

Step 10.5.2

Combine the opposite terms in $x^{2} - 4 x + 4 - 4$ .

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Step 10.5.2.1

Subtract $4$ from $4$ .

$y = x^{2} - 4 x + 0$

Step 10.5.2.2

Add $x^{2} - 4 x$ and $0$ .

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

Step 11

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

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

Step 12

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

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Step 12.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)}$ and $f^{- 1} (x) = x^{2} - 4 x$ and compare them.

Step 12.2

Find the range of $f (x) = 2 + \sqrt{- 1 (- 4 - x)}, 2 - \sqrt{- 1 (- 4 - x)}$ .

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Step 12.2.1

Find the range of $2 + \sqrt{- 1 (- 4 - x)}$ .

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Step 12.2.1.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[2, \infty)$

Step 12.2.2

Find the range of $2 - \sqrt{- 1 (- 4 - x)}$ .

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Step 12.2.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, 2]$

Step 12.2.3

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Step 12.2.3.1

The union consists of all of the elements that are contained in each interval.

$(- \infty, \infty)$

Step 12.3

Find the domain of $2 + \sqrt{- 1 (- 4 - x)}$ .

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Step 12.3.1

Set the radicand in $\sqrt{- 1 (- 4 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$- 1 (- 4 - x) \geq 0$

Step 12.3.2

Solve for $x$ .

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Step 12.3.2.1

Divide each term in $- 1 (- 4 - x) \geq 0$ by $- 1$ and simplify.

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Step 12.3.2.1.1

Divide each term in $- 1 (- 4 - x) \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 1 (- 4 - x)}{- 1} \leq \frac{0}{- 1}$

Step 12.3.2.1.2

Simplify the left side.

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Step 12.3.2.1.2.1

Dividing two negative values results in a positive value.

$\frac{- 4 - x}{1} \leq \frac{0}{- 1}$

Step 12.3.2.1.2.2

Divide $- 4 - x$ by $1$ .

$- 4 - x \leq \frac{0}{- 1}$

Step 12.3.2.1.3

Simplify the right side.

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Step 12.3.2.1.3.1

Divide $0$ by $- 1$ .

$- 4 - x \leq 0$

Step 12.3.2.2

Add $4$ to both sides of the inequality.

$- x \leq 4$

Step 12.3.2.3

Divide each term in $- x \leq 4$ by $- 1$ and simplify.

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Step 12.3.2.3.1

Divide each term in $- x \leq 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{4}{- 1}$

Step 12.3.2.3.2

Simplify the left side.

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Step 12.3.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{4}{- 1}$

Step 12.3.2.3.2.2

Divide $x$ by $1$ .

$x \geq \frac{4}{- 1}$

Step 12.3.2.3.3

Simplify the right side.

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Step 12.3.2.3.3.1

Divide $4$ by $- 1$ .

$x \geq - 4$

Step 12.3.3

The domain is all values of $x$ that make the expression defined.

$[- 4, \infty)$

Step 12.4

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

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

Step 13