Trigonometry Examples

$\frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$

Step 1

Interchange the variables.

$x = \frac{y^{2}}{8} + \frac{3 y^{2}}{2} + \frac{9 y}{2}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{y^{2}}{8} + \frac{3 y^{2}}{2} + \frac{9 y}{2} = x$ .

$\frac{y^{2}}{8} + \frac{3 y^{2}}{2} + \frac{9 y}{2} = x$

Step 2.2

Simplify $\frac{y^{2}}{8} + \frac{3 y^{2}}{2} + \frac{9 y}{2}$ .

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Step 2.2.1

To write $\frac{3 y^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{y^{2}}{8} + \frac{3 y^{2}}{2} \cdot \frac{4}{4} + \frac{9 y}{2} = x$

Step 2.2.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.2.1

Multiply $\frac{3 y^{2}}{2}$ by $\frac{4}{4}$ .

$\frac{y^{2}}{8} + \frac{3 y^{2} \cdot 4}{2 \cdot 4} + \frac{9 y}{2} = x$

Step 2.2.2.2

Multiply $2$ by $4$ .

$\frac{y^{2}}{8} + \frac{3 y^{2} \cdot 4}{8} + \frac{9 y}{2} = x$

Step 2.2.3

Combine the numerators over the common denominator.

$\frac{y^{2} + 3 y^{2} \cdot 4}{8} + \frac{9 y}{2} = x$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

Factor $y^{2}$ out of $y^{2} + 3 y^{2} \cdot 4$ .

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Step 2.2.4.1.1.1

Multiply by $1$ .

$\frac{y^{2} \cdot 1 + 3 y^{2} \cdot 4}{8} + \frac{9 y}{2} = x$

Step 2.2.4.1.1.2

Factor $y^{2}$ out of $3 y^{2} \cdot 4$ .

$\frac{y^{2} \cdot 1 + y^{2} (3 \cdot 4)}{8} + \frac{9 y}{2} = x$

Step 2.2.4.1.1.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} (3 \cdot 4)$ .

$\frac{y^{2} (1 + 3 \cdot 4)}{8} + \frac{9 y}{2} = x$

Step 2.2.4.1.2

Multiply $3$ by $4$ .

$\frac{y^{2} (1 + 12)}{8} + \frac{9 y}{2} = x$

Step 2.2.4.1.3

Add $1$ and $12$ .

$\frac{y^{2} \cdot 13}{8} + \frac{9 y}{2} = x$

Step 2.2.4.2

Move $13$ to the left of $y^{2}$ .

$\frac{13 y^{2}}{8} + \frac{9 y}{2} = x$

Step 2.3

Subtract $x$ from both sides of the equation.

$\frac{13 y^{2}}{8} + \frac{9 y}{2} - x = 0$

Step 2.4

Multiply through by the least common denominator $8$ , then simplify.

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Step 2.4.1

Apply the distributive property.

$8 (\frac{13 y^{2}}{8}) + 8 (\frac{9 y}{2}) + 8 (- x) = 0$

Step 2.4.2

Simplify.

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Step 2.4.2.1

Cancel the common factor of $8$ .

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Step 2.4.2.1.1

Cancel the common factor.

$8 (\frac{13 y^{2}}{8}) + 8 (\frac{9 y}{2}) + 8 (- x) = 0$

Step 2.4.2.1.2

Rewrite the expression.

$13 y^{2} + 8 (\frac{9 y}{2}) + 8 (- x) = 0$

Step 2.4.2.2

Cancel the common factor of $2$ .

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Step 2.4.2.2.1

Factor $2$ out of $8$ .

$13 y^{2} + 2 (4) (\frac{9 y}{2}) + 8 (- x) = 0$

Step 2.4.2.2.2

Cancel the common factor.

$13 y^{2} + 2 \cdot (4 (\frac{9 y}{2})) + 8 (- x) = 0$

Step 2.4.2.2.3

Rewrite the expression.

$13 y^{2} + 4 (9 y) + 8 (- x) = 0$

Step 2.4.2.3

Multiply $9$ by $4$ .

$13 y^{2} + 36 y + 8 (- x) = 0$

Step 2.4.2.4

Multiply $- 1$ by $8$ .

$13 y^{2} + 36 y - 8 x = 0$

Step 2.4.3

Move $36 y$ .

$13 y^{2} - 8 x + 36 y = 0$

Step 2.5

Use the quadratic formula to find the solutions.

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

Step 2.6

Substitute the values $a = 13$ , $b = 36$ , and $c = - 8 x$ into the quadratic formula and solve for $y$ .

$\frac{- 36 \pm \sqrt{36^{2} - 4 \cdot (13 \cdot (- 8 x))}}{2 \cdot 13}$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $36$ to the power of $2$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot 13 \cdot (- 8 x)}}{2 \cdot 13}$

Step 2.7.1.2

Multiply $- 4 \cdot 13 \cdot - 8$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $13$ .

$y = \frac{- 36 \pm \sqrt{1296 - 52 \cdot (- 8 x)}}{2 \cdot 13}$

Step 2.7.1.2.2

Multiply $- 52$ by $- 8$ .

$y = \frac{- 36 \pm \sqrt{1296 + 416 x}}{2 \cdot 13}$

Step 2.7.1.3

Factor $16$ out of $1296 + 416 x$ .

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Step 2.7.1.3.1

Factor $16$ out of $1296$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 416 x}}{2 \cdot 13}$

Step 2.7.1.3.2

Factor $16$ out of $416 x$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 16 (26 x)}}{2 \cdot 13}$

Step 2.7.1.3.3

Factor $16$ out of $16 (81) + 16 (26 x)$ .

$y = \frac{- 36 \pm \sqrt{16 (81 + 26 x)}}{2 \cdot 13}$

Step 2.7.1.4

Rewrite $16 (81 + 26 x)$ as ${(2^{2})}^{2} (81 + 26 x)$ .

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Step 2.7.1.4.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 36 \pm \sqrt{4^{2} (81 + 26 x)}}{2 \cdot 13}$

Step 2.7.1.4.2

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

$y = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (81 + 26 x)}}{2 \cdot 13}$

Step 2.7.1.5

Pull terms out from under the radical.

$y = \frac{- 36 \pm 2^{2} \sqrt{81 + 26 x}}{2 \cdot 13}$

Step 2.7.1.6

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

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{2 \cdot 13}$

Step 2.7.2

Multiply $2$ by $13$ .

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$

Step 2.7.3

Simplify $\frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$ .

$y = \frac{- 18 \pm 2 \sqrt{81 + 26 x}}{13}$

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Raise $36$ to the power of $2$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot 13 \cdot (- 8 x)}}{2 \cdot 13}$

Step 2.8.1.2

Multiply $- 4 \cdot 13 \cdot - 8$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $13$ .

$y = \frac{- 36 \pm \sqrt{1296 - 52 \cdot (- 8 x)}}{2 \cdot 13}$

Step 2.8.1.2.2

Multiply $- 52$ by $- 8$ .

$y = \frac{- 36 \pm \sqrt{1296 + 416 x}}{2 \cdot 13}$

Step 2.8.1.3

Factor $16$ out of $1296 + 416 x$ .

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Step 2.8.1.3.1

Factor $16$ out of $1296$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 416 x}}{2 \cdot 13}$

Step 2.8.1.3.2

Factor $16$ out of $416 x$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 16 (26 x)}}{2 \cdot 13}$

Step 2.8.1.3.3

Factor $16$ out of $16 (81) + 16 (26 x)$ .

$y = \frac{- 36 \pm \sqrt{16 (81 + 26 x)}}{2 \cdot 13}$

Step 2.8.1.4

Rewrite $16 (81 + 26 x)$ as ${(2^{2})}^{2} (81 + 26 x)$ .

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Step 2.8.1.4.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 36 \pm \sqrt{4^{2} (81 + 26 x)}}{2 \cdot 13}$

Step 2.8.1.4.2

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

$y = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (81 + 26 x)}}{2 \cdot 13}$

Step 2.8.1.5

Pull terms out from under the radical.

$y = \frac{- 36 \pm 2^{2} \sqrt{81 + 26 x}}{2 \cdot 13}$

Step 2.8.1.6

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

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{2 \cdot 13}$

Step 2.8.2

Multiply $2$ by $13$ .

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$

Step 2.8.3

Simplify $\frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$ .

$y = \frac{- 18 \pm 2 \sqrt{81 + 26 x}}{13}$

Step 2.8.4

Change the $\pm$ to $+$ .

$y = \frac{- 18 + 2 \sqrt{81 + 26 x}}{13}$

Step 2.8.5

Factor $2$ out of $- 18 + 2 \sqrt{81 + 26 x}$ .

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Step 2.8.5.1

Factor $2$ out of $- 18$ .

$y = \frac{2 \cdot - 9 + 2 \sqrt{81 + 26 x}}{13}$

Step 2.8.5.2

Factor $2$ out of $2 \cdot - 9 + 2 \sqrt{81 + 26 x}$ .

$y = \frac{2 (- 9 + \sqrt{81 + 26 x})}{13}$

Step 2.8.6

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

$y = \frac{2 (- 1 \cdot 9 + \sqrt{81 + 26 x})}{13}$

Step 2.8.7

Factor $- 1$ out of $\sqrt{81 + 26 x}$ .

$y = \frac{2 (- 1 \cdot 9 - 1 (- \sqrt{81 + 26 x}))}{13}$

Step 2.8.8

Factor $- 1$ out of $- 1 (9) - 1 (- \sqrt{81 + 26 x})$ .

$y = \frac{2 (- 1 (9 - \sqrt{81 + 26 x}))}{13}$

Step 2.8.9

Move the negative in front of the fraction.

$y = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}$

Step 2.9

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

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Step 2.9.1

Simplify the numerator.

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Step 2.9.1.1

Raise $36$ to the power of $2$ .

$y = \frac{- 36 \pm \sqrt{1296 - 4 \cdot 13 \cdot (- 8 x)}}{2 \cdot 13}$

Step 2.9.1.2

Multiply $- 4 \cdot 13 \cdot - 8$ .

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Step 2.9.1.2.1

Multiply $- 4$ by $13$ .

$y = \frac{- 36 \pm \sqrt{1296 - 52 \cdot (- 8 x)}}{2 \cdot 13}$

Step 2.9.1.2.2

Multiply $- 52$ by $- 8$ .

$y = \frac{- 36 \pm \sqrt{1296 + 416 x}}{2 \cdot 13}$

Step 2.9.1.3

Factor $16$ out of $1296 + 416 x$ .

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Step 2.9.1.3.1

Factor $16$ out of $1296$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 416 x}}{2 \cdot 13}$

Step 2.9.1.3.2

Factor $16$ out of $416 x$ .

$y = \frac{- 36 \pm \sqrt{16 (81) + 16 (26 x)}}{2 \cdot 13}$

Step 2.9.1.3.3

Factor $16$ out of $16 (81) + 16 (26 x)$ .

$y = \frac{- 36 \pm \sqrt{16 (81 + 26 x)}}{2 \cdot 13}$

Step 2.9.1.4

Rewrite $16 (81 + 26 x)$ as ${(2^{2})}^{2} (81 + 26 x)$ .

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Step 2.9.1.4.1

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 36 \pm \sqrt{4^{2} (81 + 26 x)}}{2 \cdot 13}$

Step 2.9.1.4.2

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

$y = \frac{- 36 \pm \sqrt{{(2^{2})}^{2} (81 + 26 x)}}{2 \cdot 13}$

Step 2.9.1.5

Pull terms out from under the radical.

$y = \frac{- 36 \pm 2^{2} \sqrt{81 + 26 x}}{2 \cdot 13}$

Step 2.9.1.6

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

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{2 \cdot 13}$

Step 2.9.2

Multiply $2$ by $13$ .

$y = \frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$

Step 2.9.3

Simplify $\frac{- 36 \pm 4 \sqrt{81 + 26 x}}{26}$ .

$y = \frac{- 18 \pm 2 \sqrt{81 + 26 x}}{13}$

Step 2.9.4

Change the $\pm$ to $-$ .

$y = \frac{- 18 - 2 \sqrt{81 + 26 x}}{13}$

Step 2.9.5

Factor $- 2$ out of $- 18 - 2 \sqrt{81 + 26 x}$ .

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Step 2.9.5.1

Reorder $81$ and $26 x$ .

$y = \frac{- 18 - 2 \sqrt{26 x + 81}}{13}$

Step 2.9.5.2

Factor $- 2$ out of $- 18$ .

$y = \frac{- 2 \cdot 9 - 2 \sqrt{26 x + 81}}{13}$

Step 2.9.5.3

Factor $- 2$ out of $- 2 \sqrt{26 x + 81}$ .

$y = \frac{- 2 \cdot 9 - 2 \sqrt{26 x + 81}}{13}$

Step 2.9.5.4

Factor $- 2$ out of $- 2 (9) - 2 (\sqrt{26 x + 81})$ .

$y = \frac{- 2 (9 + \sqrt{26 x + 81})}{13}$

Step 2.9.6

Move the negative in front of the fraction.

$y = - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

Step 2.10

The final answer is the combination of both solutions.

$y = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}$

$y = - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

$y = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}$

$y = - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

Step 3

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

$f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

Step 4

Verify if $f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$ is the inverse of $f (x) = \frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$ .

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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 $f (x) = \frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$ and $f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$ and compare them.

Step 4.2

Find the range of $f (x) = \frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$ .

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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:

$[- \frac{81}{26}, \infty)$

Step 4.3

Find the domain of $- \frac{2 (9 - \sqrt{81 + 26 x})}{13}$ .

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Step 4.3.1

Set the radicand in $\sqrt{81 + 26 x}$ greater than or equal to $0$ to find where the expression is defined.

$81 + 26 x \geq 0$

Step 4.3.2

Solve for $x$ .

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Step 4.3.2.1

Subtract $81$ from both sides of the inequality.

$26 x \geq - 81$

Step 4.3.2.2

Divide each term in $26 x \geq - 81$ by $26$ and simplify.

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Step 4.3.2.2.1

Divide each term in $26 x \geq - 81$ by $26$ .

$\frac{26 x}{26} \geq \frac{- 81}{26}$

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 $26$ .

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Step 4.3.2.2.2.1.1

Cancel the common factor.

$\frac{26 x}{26} \geq \frac{- 81}{26}$

Step 4.3.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 81}{26}$

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.

$x \geq - \frac{81}{26}$

Step 4.3.3

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

$[- \frac{81}{26}, \infty)$

Step 4.4

Find the domain of $f (x) = \frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$ .

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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 $f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$ is the range of $f (x) = \frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$ and the range of $f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$ is the domain of $f (x) = \frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$ , then $f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$ is the inverse of $f (x) = \frac{x^{2}}{8} + \frac{3 x^{2}}{2} + \frac{9 x}{2}$ .

$f^{- 1} (x) = - \frac{2 (9 - \sqrt{81 + 26 x})}{13}, - \frac{2 (9 + \sqrt{26 x + 81})}{13}$

Step 5