Trigonometry Examples

$2 x^{2} + 18 x + 40$

Step 1

Interchange the variables.

$x = 2 y^{2} + 18 y + 40$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $2 y^{2} + 18 y + 40 = x$ .

$2 y^{2} + 18 y + 40 = x$

Step 2.2

Subtract $x$ from both sides of the equation.

$2 y^{2} + 18 y + 40 - x = 0$

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 $a = 2$ , $b = 18$ , and $c = 40 - x$ into the quadratic formula and solve for $y$ .

$\frac{- 18 \pm \sqrt{18^{2} - 4 \cdot (2 \cdot (40 - x))}}{2 \cdot 2}$

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 $18$ to the power of $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 4 \cdot 2 \cdot (40 - x)}}{2 \cdot 2}$

Step 2.5.1.2

Multiply $- 4$ by $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 8 \cdot (40 - x)}}{2 \cdot 2}$

Step 2.5.1.3

Apply the distributive property.

$y = \frac{- 18 \pm \sqrt{324 - 8 \cdot 40 - 8 (- x)}}{2 \cdot 2}$

Step 2.5.1.4

Multiply $- 8$ by $40$ .

$y = \frac{- 18 \pm \sqrt{324 - 320 - 8 (- x)}}{2 \cdot 2}$

Step 2.5.1.5

Multiply $- 1$ by $- 8$ .

$y = \frac{- 18 \pm \sqrt{324 - 320 + 8 x}}{2 \cdot 2}$

Step 2.5.1.6

Subtract $320$ from $324$ .

$y = \frac{- 18 \pm \sqrt{4 + 8 x}}{2 \cdot 2}$

Step 2.5.1.7

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

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Step 2.5.1.7.1

Factor $4$ out of $4$ .

$y = \frac{- 18 \pm \sqrt{4 (1) + 8 x}}{2 \cdot 2}$

Step 2.5.1.7.2

Factor $4$ out of $8 x$ .

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

Step 2.5.1.7.3

Factor $4$ out of $4 (1) + 4 (2 x)$ .

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

Step 2.5.1.8

Rewrite $4 (1 + 2 x)$ as $2^{2} (1^{2} + 2 x)$ .

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Step 2.5.1.8.1

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

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

Step 2.5.1.8.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.5.1.9

Pull terms out from under the radical.

$y = \frac{- 18 \pm 2 \sqrt{1^{2} + 2 x}}{2 \cdot 2}$

Step 2.5.1.10

One to any power is one.

$y = \frac{- 18 \pm 2 \sqrt{1 + 2 x}}{2 \cdot 2}$

Step 2.5.2

Multiply $2$ by $2$ .

$y = \frac{- 18 \pm 2 \sqrt{1 + 2 x}}{4}$

Step 2.5.3

Simplify $\frac{- 18 \pm 2 \sqrt{1 + 2 x}}{4}$ .

$y = \frac{- 9 \pm \sqrt{1 + 2 x}}{2}$

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 $18$ to the power of $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 4 \cdot 2 \cdot (40 - x)}}{2 \cdot 2}$

Step 2.6.1.2

Multiply $- 4$ by $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 8 \cdot (40 - x)}}{2 \cdot 2}$

Step 2.6.1.3

Apply the distributive property.

$y = \frac{- 18 \pm \sqrt{324 - 8 \cdot 40 - 8 (- x)}}{2 \cdot 2}$

Step 2.6.1.4

Multiply $- 8$ by $40$ .

$y = \frac{- 18 \pm \sqrt{324 - 320 - 8 (- x)}}{2 \cdot 2}$

Step 2.6.1.5

Multiply $- 1$ by $- 8$ .

$y = \frac{- 18 \pm \sqrt{324 - 320 + 8 x}}{2 \cdot 2}$

Step 2.6.1.6

Subtract $320$ from $324$ .

$y = \frac{- 18 \pm \sqrt{4 + 8 x}}{2 \cdot 2}$

Step 2.6.1.7

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

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Step 2.6.1.7.1

Factor $4$ out of $4$ .

$y = \frac{- 18 \pm \sqrt{4 (1) + 8 x}}{2 \cdot 2}$

Step 2.6.1.7.2

Factor $4$ out of $8 x$ .

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

Step 2.6.1.7.3

Factor $4$ out of $4 (1) + 4 (2 x)$ .

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

Step 2.6.1.8

Rewrite $4 (1 + 2 x)$ as $2^{2} (1^{2} + 2 x)$ .

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Step 2.6.1.8.1

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

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

Step 2.6.1.8.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.6.1.9

Pull terms out from under the radical.

$y = \frac{- 18 \pm 2 \sqrt{1^{2} + 2 x}}{2 \cdot 2}$

Step 2.6.1.10

One to any power is one.

$y = \frac{- 18 \pm 2 \sqrt{1 + 2 x}}{2 \cdot 2}$

Step 2.6.2

Multiply $2$ by $2$ .

$y = \frac{- 18 \pm 2 \sqrt{1 + 2 x}}{4}$

Step 2.6.3

Simplify $\frac{- 18 \pm 2 \sqrt{1 + 2 x}}{4}$ .

$y = \frac{- 9 \pm \sqrt{1 + 2 x}}{2}$

Step 2.6.4

Change the $\pm$ to $+$ .

$y = \frac{- 9 + \sqrt{1 + 2 x}}{2}$

Step 2.6.5

Rewrite $- 9$ as $- 1 (9)$ .

$y = \frac{- 1 \cdot 9 + \sqrt{1 + 2 x}}{2}$

Step 2.6.6

Factor $- 1$ out of $\sqrt{1 + 2 x}$ .

$y = \frac{- 1 \cdot 9 - 1 (- \sqrt{1 + 2 x})}{2}$

Step 2.6.7

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{1 + 2 x})$ .

$y = \frac{- 1 (9 - \sqrt{1 + 2 x})}{2}$

Step 2.6.8

Move the negative in front of the fraction.

$y = - \frac{9 - \sqrt{1 + 2 x}}{2}$

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 $18$ to the power of $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 4 \cdot 2 \cdot (40 - x)}}{2 \cdot 2}$

Step 2.7.1.2

Multiply $- 4$ by $2$ .

$y = \frac{- 18 \pm \sqrt{324 - 8 \cdot (40 - x)}}{2 \cdot 2}$

Step 2.7.1.3

Apply the distributive property.

$y = \frac{- 18 \pm \sqrt{324 - 8 \cdot 40 - 8 (- x)}}{2 \cdot 2}$

Step 2.7.1.4

Multiply $- 8$ by $40$ .

$y = \frac{- 18 \pm \sqrt{324 - 320 - 8 (- x)}}{2 \cdot 2}$

Step 2.7.1.5

Multiply $- 1$ by $- 8$ .

$y = \frac{- 18 \pm \sqrt{324 - 320 + 8 x}}{2 \cdot 2}$

Step 2.7.1.6

Subtract $320$ from $324$ .

$y = \frac{- 18 \pm \sqrt{4 + 8 x}}{2 \cdot 2}$

Step 2.7.1.7

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

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Step 2.7.1.7.1

Factor $4$ out of $4$ .

$y = \frac{- 18 \pm \sqrt{4 (1) + 8 x}}{2 \cdot 2}$

Step 2.7.1.7.2

Factor $4$ out of $8 x$ .

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

Step 2.7.1.7.3

Factor $4$ out of $4 (1) + 4 (2 x)$ .

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

Step 2.7.1.8

Rewrite $4 (1 + 2 x)$ as $2^{2} (1^{2} + 2 x)$ .

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Step 2.7.1.8.1

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

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

Step 2.7.1.8.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.7.1.9

Pull terms out from under the radical.

$y = \frac{- 18 \pm 2 \sqrt{1^{2} + 2 x}}{2 \cdot 2}$

Step 2.7.1.10

One to any power is one.

$y = \frac{- 18 \pm 2 \sqrt{1 + 2 x}}{2 \cdot 2}$

Step 2.7.2

Multiply $2$ by $2$ .

$y = \frac{- 18 \pm 2 \sqrt{1 + 2 x}}{4}$

Step 2.7.3

Simplify $\frac{- 18 \pm 2 \sqrt{1 + 2 x}}{4}$ .

$y = \frac{- 9 \pm \sqrt{1 + 2 x}}{2}$

Step 2.7.4

Change the $\pm$ to $-$ .

$y = \frac{- 9 - \sqrt{1 + 2 x}}{2}$

Step 2.7.5

Factor $- 1$ out of $- 9 - \sqrt{1 + 2 x}$ .

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Step 2.7.5.1

Reorder $1$ and $2 x$ .

$y = \frac{- 9 - \sqrt{2 x + 1}}{2}$

Step 2.7.5.2

Rewrite $- 9$ as $- 1 (9)$ .

$y = \frac{- 1 \cdot 9 - \sqrt{2 x + 1}}{2}$

Step 2.7.5.3

Factor $- 1$ out of $- \sqrt{2 x + 1}$ .

$y = \frac{- 1 \cdot 9 - (\sqrt{2 x + 1})}{2}$

Step 2.7.5.4

Factor $- 1$ out of $- 1 (9) - (\sqrt{2 x + 1})$ .

$y = \frac{- 1 (9 + \sqrt{2 x + 1})}{2}$

Step 2.7.5.5

Rewrite $- 1 (9 + \sqrt{2 x + 1})$ as $- (9 + \sqrt{2 x + 1})$ .

$y = \frac{- (9 + \sqrt{2 x + 1})}{2}$

Step 2.7.6

Move the negative in front of the fraction.

$y = - \frac{9 + \sqrt{2 x + 1}}{2}$

Step 2.8

The final answer is the combination of both solutions.

$y = - \frac{9 - \sqrt{1 + 2 x}}{2}$

$y = - \frac{9 + \sqrt{2 x + 1}}{2}$

$y = - \frac{9 - \sqrt{1 + 2 x}}{2}$

$y = - \frac{9 + \sqrt{2 x + 1}}{2}$

Step 3

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

$f^{- 1} (x) = - \frac{9 - \sqrt{1 + 2 x}}{2}, - \frac{9 + \sqrt{2 x + 1}}{2}$

Step 4

Verify if $f^{- 1} (x) = - \frac{9 - \sqrt{1 + 2 x}}{2}, - \frac{9 + \sqrt{2 x + 1}}{2}$ is the inverse of $f (x) = 2 x^{2} + 18 x + 40$ .

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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 $f (x) = 2 x^{2} + 18 x + 40$ and $f^{- 1} (x) = - \frac{9 - \sqrt{1 + 2 x}}{2}, - \frac{9 + \sqrt{2 x + 1}}{2}$ and compare them.

Step 4.2

Find the range of $f (x) = 2 x^{2} + 18 x + 40$ .

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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:

$[- \frac{1}{2}, \infty)$

Step 4.3

Find the domain of $- \frac{9 - \sqrt{1 + 2 x}}{2}$ .

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Step 4.3.1

Set the radicand in $\sqrt{1 + 2 x}$ greater than or equal to $0$ to find where the expression is defined.

$1 + 2 x \geq 0$

Step 4.3.2

Solve for $x$ .

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Step 4.3.2.1

Subtract $1$ from both sides of the inequality.

$2 x \geq - 1$

Step 4.3.2.2

Divide each term in $2 x \geq - 1$ by $2$ and simplify.

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Step 4.3.2.2.1

Divide each term in $2 x \geq - 1$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 1}{2}$

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 1}{2}$

Step 4.3.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 1}{2}$

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{1}{2}$

Step 4.3.3

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

$[- \frac{1}{2}, \infty)$

Step 4.4

Find the domain of $f (x) = 2 x^{2} + 18 x + 40$ .

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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 $f^{- 1} (x) = - \frac{9 - \sqrt{1 + 2 x}}{2}, - \frac{9 + \sqrt{2 x + 1}}{2}$ is the range of $f (x) = 2 x^{2} + 18 x + 40$ and the range of $f^{- 1} (x) = - \frac{9 - \sqrt{1 + 2 x}}{2}, - \frac{9 + \sqrt{2 x + 1}}{2}$ is the domain of $f (x) = 2 x^{2} + 18 x + 40$ , then $f^{- 1} (x) = - \frac{9 - \sqrt{1 + 2 x}}{2}, - \frac{9 + \sqrt{2 x + 1}}{2}$ is the inverse of $f (x) = 2 x^{2} + 18 x + 40$ .

$f^{- 1} (x) = - \frac{9 - \sqrt{1 + 2 x}}{2}, - \frac{9 + \sqrt{2 x + 1}}{2}$

Step 5