Precalculus Examples

$y = - x^{2} - 10 x - 18$

Step 1

Interchange the variables.

$x = - y^{2} - 10 y - 18$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- y^{2} - 10 y - 18 = x$ .

$- y^{2} - 10 y - 18 = x$

Step 2.2

Subtract $x$ from both sides of the equation.

$- y^{2} - 10 y - 18 - x = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = - 1$ , $b = - 10$ , and $c = - 18 - x$ into the quadratic formula and solve for $y$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (- 1 \cdot (- 18 - x))}}{2 \cdot - 1}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot (- 18 - x)}}{2 \cdot - 1}$

Step 2.5.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{10 \pm \sqrt{100 + 4 \cdot (- 18 - x)}}{2 \cdot - 1}$

Step 2.5.1.3

Apply the distributive property.

$y = \frac{10 \pm \sqrt{100 + 4 \cdot - 18 + 4 (- x)}}{2 \cdot - 1}$

Step 2.5.1.4

Multiply $4$ by $- 18$ .

$y = \frac{10 \pm \sqrt{100 - 72 + 4 (- x)}}{2 \cdot - 1}$

Step 2.5.1.5

Multiply $- 1$ by $4$ .

$y = \frac{10 \pm \sqrt{100 - 72 - 4 x}}{2 \cdot - 1}$

Step 2.5.1.6

Subtract $72$ from $100$ .

$y = \frac{10 \pm \sqrt{28 - 4 x}}{2 \cdot - 1}$

Step 2.5.1.7

Factor $4$ out of $28 - 4 x$ .

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Step 2.5.1.7.1

Factor $4$ out of $28$ .

$y = \frac{10 \pm \sqrt{4 (7) - 4 x}}{2 \cdot - 1}$

Step 2.5.1.7.2

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

$y = \frac{10 \pm \sqrt{4 (7) + 4 (- x)}}{2 \cdot - 1}$

Step 2.5.1.7.3

Factor $4$ out of $4 (7) + 4 (- x)$ .

$y = \frac{10 \pm \sqrt{4 (7 - x)}}{2 \cdot - 1}$

Step 2.5.1.8

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

$y = \frac{10 \pm \sqrt{2^{2} (7 - x)}}{2 \cdot - 1}$

Step 2.5.1.9

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{7 - x}}{2 \cdot - 1}$

Step 2.5.2

Multiply $2$ by $- 1$ .

$y = \frac{10 \pm 2 \sqrt{7 - x}}{- 2}$

Step 2.5.3

Simplify $\frac{10 \pm 2 \sqrt{7 - x}}{- 2}$ .

$y = \frac{5 \pm \sqrt{7 - x}}{- 1}$

Step 2.5.4

Move the negative one from the denominator of $\frac{5 \pm \sqrt{7 - x}}{- 1}$ .

$y = - 1 \cdot (5 \pm \sqrt{7 - x})$

Step 2.5.5

Rewrite $- 1 \cdot (5 \pm \sqrt{7 - x})$ as $- (5 \pm \sqrt{7 - x})$ .

$y = - (5 \pm \sqrt{7 - x})$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot (- 18 - x)}}{2 \cdot - 1}$

Step 2.6.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{10 \pm \sqrt{100 + 4 \cdot (- 18 - x)}}{2 \cdot - 1}$

Step 2.6.1.3

Apply the distributive property.

$y = \frac{10 \pm \sqrt{100 + 4 \cdot - 18 + 4 (- x)}}{2 \cdot - 1}$

Step 2.6.1.4

Multiply $4$ by $- 18$ .

$y = \frac{10 \pm \sqrt{100 - 72 + 4 (- x)}}{2 \cdot - 1}$

Step 2.6.1.5

Multiply $- 1$ by $4$ .

$y = \frac{10 \pm \sqrt{100 - 72 - 4 x}}{2 \cdot - 1}$

Step 2.6.1.6

Subtract $72$ from $100$ .

$y = \frac{10 \pm \sqrt{28 - 4 x}}{2 \cdot - 1}$

Step 2.6.1.7

Factor $4$ out of $28 - 4 x$ .

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Step 2.6.1.7.1

Factor $4$ out of $28$ .

$y = \frac{10 \pm \sqrt{4 (7) - 4 x}}{2 \cdot - 1}$

Step 2.6.1.7.2

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

$y = \frac{10 \pm \sqrt{4 (7) + 4 (- x)}}{2 \cdot - 1}$

Step 2.6.1.7.3

Factor $4$ out of $4 (7) + 4 (- x)$ .

$y = \frac{10 \pm \sqrt{4 (7 - x)}}{2 \cdot - 1}$

Step 2.6.1.8

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

$y = \frac{10 \pm \sqrt{2^{2} (7 - x)}}{2 \cdot - 1}$

Step 2.6.1.9

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{7 - x}}{2 \cdot - 1}$

Step 2.6.2

Multiply $2$ by $- 1$ .

$y = \frac{10 \pm 2 \sqrt{7 - x}}{- 2}$

Step 2.6.3

Simplify $\frac{10 \pm 2 \sqrt{7 - x}}{- 2}$ .

$y = \frac{5 \pm \sqrt{7 - x}}{- 1}$

Step 2.6.4

Move the negative one from the denominator of $\frac{5 \pm \sqrt{7 - x}}{- 1}$ .

$y = - 1 \cdot (5 \pm \sqrt{7 - x})$

Step 2.6.5

Rewrite $- 1 \cdot (5 \pm \sqrt{7 - x})$ as $- (5 \pm \sqrt{7 - x})$ .

$y = - (5 \pm \sqrt{7 - x})$

Step 2.6.6

Change the $\pm$ to $+$ .

$y = - (5 + \sqrt{7 - x})$

Step 2.6.7

Apply the distributive property.

$y = - 1 \cdot 5 - \sqrt{7 - x}$

Step 2.6.8

Multiply $- 1$ by $5$ .

$y = - 5 - \sqrt{7 - x}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot - 1 \cdot (- 18 - x)}}{2 \cdot - 1}$

Step 2.7.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{10 \pm \sqrt{100 + 4 \cdot (- 18 - x)}}{2 \cdot - 1}$

Step 2.7.1.3

Apply the distributive property.

$y = \frac{10 \pm \sqrt{100 + 4 \cdot - 18 + 4 (- x)}}{2 \cdot - 1}$

Step 2.7.1.4

Multiply $4$ by $- 18$ .

$y = \frac{10 \pm \sqrt{100 - 72 + 4 (- x)}}{2 \cdot - 1}$

Step 2.7.1.5

Multiply $- 1$ by $4$ .

$y = \frac{10 \pm \sqrt{100 - 72 - 4 x}}{2 \cdot - 1}$

Step 2.7.1.6

Subtract $72$ from $100$ .

$y = \frac{10 \pm \sqrt{28 - 4 x}}{2 \cdot - 1}$

Step 2.7.1.7

Factor $4$ out of $28 - 4 x$ .

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Step 2.7.1.7.1

Factor $4$ out of $28$ .

$y = \frac{10 \pm \sqrt{4 (7) - 4 x}}{2 \cdot - 1}$

Step 2.7.1.7.2

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

$y = \frac{10 \pm \sqrt{4 (7) + 4 (- x)}}{2 \cdot - 1}$

Step 2.7.1.7.3

Factor $4$ out of $4 (7) + 4 (- x)$ .

$y = \frac{10 \pm \sqrt{4 (7 - x)}}{2 \cdot - 1}$

Step 2.7.1.8

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

$y = \frac{10 \pm \sqrt{2^{2} (7 - x)}}{2 \cdot - 1}$

Step 2.7.1.9

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{7 - x}}{2 \cdot - 1}$

Step 2.7.2

Multiply $2$ by $- 1$ .

$y = \frac{10 \pm 2 \sqrt{7 - x}}{- 2}$

Step 2.7.3

Simplify $\frac{10 \pm 2 \sqrt{7 - x}}{- 2}$ .

$y = \frac{5 \pm \sqrt{7 - x}}{- 1}$

Step 2.7.4

Move the negative one from the denominator of $\frac{5 \pm \sqrt{7 - x}}{- 1}$ .

$y = - 1 \cdot (5 \pm \sqrt{7 - x})$

Step 2.7.5

Rewrite $- 1 \cdot (5 \pm \sqrt{7 - x})$ as $- (5 \pm \sqrt{7 - x})$ .

$y = - (5 \pm \sqrt{7 - x})$

Step 2.7.6

Change the $\pm$ to $-$ .

$y = - (5 - \sqrt{7 - x})$

Step 2.7.7

Apply the distributive property.

$y = - 1 \cdot 5 + \sqrt{7 - x}$

Step 2.7.8

Multiply $- 1$ by $5$ .

$y = - 5 + \sqrt{7 - x}$

Step 2.7.9

Multiply $- - \sqrt{7 - x}$ .

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Step 2.7.9.1

Multiply $- 1$ by $- 1$ .

$y = - 5 + 1 \sqrt{7 - x}$

Step 2.7.9.2

Multiply $\sqrt{7 - x}$ by $1$ .

$y = - 5 + \sqrt{7 - x}$

Step 2.8

The final answer is the combination of both solutions.

$y = - 5 - \sqrt{7 - x}$

$y = - 5 + \sqrt{7 - x}$

$y = - 5 - \sqrt{7 - x}$

$y = - 5 + \sqrt{7 - x}$

Step 3

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

$f^{- 1} (x) = - 5 - \sqrt{7 - x}, - 5 + \sqrt{7 - x}$

Step 4

Verify if $f^{- 1} (x) = - 5 - \sqrt{7 - x}, - 5 + \sqrt{7 - x}$ is the inverse of $f (x) = - x^{2} - 10 x - 18$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = - x^{2} - 10 x - 18$ and $f^{- 1} (x) = - 5 - \sqrt{7 - x}, - 5 + \sqrt{7 - x}$ and compare them.

Step 4.2

Find the range of $f (x) = - x^{2} - 10 x - 18$ .

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Step 4.2.1

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

Interval Notation:

$(- \infty, 7]$

Step 4.3

Find the domain of $- 5 - \sqrt{7 - x}$ .

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Step 4.3.1

Set the radicand in $\sqrt{7 - x}$ greater than or equal to $0$ to find where the expression is defined.

$7 - x \geq 0$

Step 4.3.2

Solve for $x$ .

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Step 4.3.2.1

Subtract $7$ from both sides of the inequality.

$- x \geq - 7$

Step 4.3.2.2

Divide each term in $- x \geq - 7$ by $- 1$ and simplify.

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Step 4.3.2.2.1

Divide each term in $- x \geq - 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 7}{- 1}$

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 7}{- 1}$

Step 4.3.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 7}{- 1}$

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Divide $- 7$ by $- 1$ .

$x \leq 7$

Step 4.3.3

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

$(- \infty, 7]$

Step 4.4

Find the domain of $f (x) = - x^{2} - 10 x - 18$ .

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Step 4.4.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Step 4.5

Since the domain of $f^{- 1} (x) = - 5 - \sqrt{7 - x}, - 5 + \sqrt{7 - x}$ is the range of $f (x) = - x^{2} - 10 x - 18$ and the range of $f^{- 1} (x) = - 5 - \sqrt{7 - x}, - 5 + \sqrt{7 - x}$ is the domain of $f (x) = - x^{2} - 10 x - 18$ , then $f^{- 1} (x) = - 5 - \sqrt{7 - x}, - 5 + \sqrt{7 - x}$ is the inverse of $f (x) = - x^{2} - 10 x - 18$ .

$f^{- 1} (x) = - 5 - \sqrt{7 - x}, - 5 + \sqrt{7 - x}$

Step 5