Trigonometry Examples

$y = 6 x^{2} - 9 x - 1$

Step 1

Interchange the variables.

$x = 6 y^{2} - 9 y - 1$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $6 y^{2} - 9 y - 1 = x$ .

$6 y^{2} - 9 y - 1 = x$

Step 2.2

Subtract $x$ from both sides of the equation.

$6 y^{2} - 9 y - 1 - x = 0$

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 $a = 6$ , $b = - 9$ , and $c = - 1 - x$ into the quadratic formula and solve for $y$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (6 \cdot (- 1 - x))}}{2 \cdot 6}$

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 $- 9$ to the power of $2$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 6 \cdot (- 1 - x)}}{2 \cdot 6}$

Step 2.5.1.2

Multiply $- 4$ by $6$ .

$y = \frac{9 \pm \sqrt{81 - 24 \cdot (- 1 - x)}}{2 \cdot 6}$

Step 2.5.1.3

Apply the distributive property.

$y = \frac{9 \pm \sqrt{81 - 24 \cdot - 1 - 24 (- x)}}{2 \cdot 6}$

Step 2.5.1.4

Multiply $- 24$ by $- 1$ .

$y = \frac{9 \pm \sqrt{81 + 24 - 24 (- x)}}{2 \cdot 6}$

Step 2.5.1.5

Multiply $- 1$ by $- 24$ .

$y = \frac{9 \pm \sqrt{81 + 24 + 24 x}}{2 \cdot 6}$

Step 2.5.1.6

Add $81$ and $24$ .

$y = \frac{9 \pm \sqrt{105 + 24 x}}{2 \cdot 6}$

Step 2.5.1.7

Factor $3$ out of $105 + 24 x$ .

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Step 2.5.1.7.1

Factor $3$ out of $105$ .

$y = \frac{9 \pm \sqrt{3 (35) + 24 x}}{2 \cdot 6}$

Step 2.5.1.7.2

Factor $3$ out of $24 x$ .

$y = \frac{9 \pm \sqrt{3 (35) + 3 (8 x)}}{2 \cdot 6}$

Step 2.5.1.7.3

Factor $3$ out of $3 (35) + 3 (8 x)$ .

$y = \frac{9 \pm \sqrt{3 (35 + 8 x)}}{2 \cdot 6}$

Step 2.5.2

Multiply $2$ by $6$ .

$y = \frac{9 \pm \sqrt{3 (35 + 8 x)}}{12}$

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 $- 9$ to the power of $2$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 6 \cdot (- 1 - x)}}{2 \cdot 6}$

Step 2.6.1.2

Multiply $- 4$ by $6$ .

$y = \frac{9 \pm \sqrt{81 - 24 \cdot (- 1 - x)}}{2 \cdot 6}$

Step 2.6.1.3

Apply the distributive property.

$y = \frac{9 \pm \sqrt{81 - 24 \cdot - 1 - 24 (- x)}}{2 \cdot 6}$

Step 2.6.1.4

Multiply $- 24$ by $- 1$ .

$y = \frac{9 \pm \sqrt{81 + 24 - 24 (- x)}}{2 \cdot 6}$

Step 2.6.1.5

Multiply $- 1$ by $- 24$ .

$y = \frac{9 \pm \sqrt{81 + 24 + 24 x}}{2 \cdot 6}$

Step 2.6.1.6

Add $81$ and $24$ .

$y = \frac{9 \pm \sqrt{105 + 24 x}}{2 \cdot 6}$

Step 2.6.1.7

Factor $3$ out of $105 + 24 x$ .

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Step 2.6.1.7.1

Factor $3$ out of $105$ .

$y = \frac{9 \pm \sqrt{3 (35) + 24 x}}{2 \cdot 6}$

Step 2.6.1.7.2

Factor $3$ out of $24 x$ .

$y = \frac{9 \pm \sqrt{3 (35) + 3 (8 x)}}{2 \cdot 6}$

Step 2.6.1.7.3

Factor $3$ out of $3 (35) + 3 (8 x)$ .

$y = \frac{9 \pm \sqrt{3 (35 + 8 x)}}{2 \cdot 6}$

Step 2.6.2

Multiply $2$ by $6$ .

$y = \frac{9 \pm \sqrt{3 (35 + 8 x)}}{12}$

Step 2.6.3

Change the $\pm$ to $+$ .

$y = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}$

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 $- 9$ to the power of $2$ .

$y = \frac{9 \pm \sqrt{81 - 4 \cdot 6 \cdot (- 1 - x)}}{2 \cdot 6}$

Step 2.7.1.2

Multiply $- 4$ by $6$ .

$y = \frac{9 \pm \sqrt{81 - 24 \cdot (- 1 - x)}}{2 \cdot 6}$

Step 2.7.1.3

Apply the distributive property.

$y = \frac{9 \pm \sqrt{81 - 24 \cdot - 1 - 24 (- x)}}{2 \cdot 6}$

Step 2.7.1.4

Multiply $- 24$ by $- 1$ .

$y = \frac{9 \pm \sqrt{81 + 24 - 24 (- x)}}{2 \cdot 6}$

Step 2.7.1.5

Multiply $- 1$ by $- 24$ .

$y = \frac{9 \pm \sqrt{81 + 24 + 24 x}}{2 \cdot 6}$

Step 2.7.1.6

Add $81$ and $24$ .

$y = \frac{9 \pm \sqrt{105 + 24 x}}{2 \cdot 6}$

Step 2.7.1.7

Factor $3$ out of $105 + 24 x$ .

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Step 2.7.1.7.1

Factor $3$ out of $105$ .

$y = \frac{9 \pm \sqrt{3 (35) + 24 x}}{2 \cdot 6}$

Step 2.7.1.7.2

Factor $3$ out of $24 x$ .

$y = \frac{9 \pm \sqrt{3 (35) + 3 (8 x)}}{2 \cdot 6}$

Step 2.7.1.7.3

Factor $3$ out of $3 (35) + 3 (8 x)$ .

$y = \frac{9 \pm \sqrt{3 (35 + 8 x)}}{2 \cdot 6}$

Step 2.7.2

Multiply $2$ by $6$ .

$y = \frac{9 \pm \sqrt{3 (35 + 8 x)}}{12}$

Step 2.7.3

Change the $\pm$ to $-$ .

$y = \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$

Step 2.8

The final answer is the combination of both solutions.

$y = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}$

$y = \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$

$y = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}$

$y = \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$

Step 3

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

$f^{- 1} (x) = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}, \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$

Step 4

Verify if $f^{- 1} (x) = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}, \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$ is the inverse of $f (x) = 6 x^{2} - 9 x - 1$ .

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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 $f (x) = 6 x^{2} - 9 x - 1$ and $f^{- 1} (x) = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}, \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$ and compare them.

Step 4.2

Find the range of $f (x) = 6 x^{2} - 9 x - 1$ .

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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:

$[- \frac{35}{8}, \infty)$

Step 4.3

Find the domain of $\frac{9 + \sqrt{3 (35 + 8 x)}}{12}$ .

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Step 4.3.1

Set the radicand in $\sqrt{3 (35 + 8 x)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (35 + 8 x) \geq 0$

Step 4.3.2

Solve for $x$ .

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Step 4.3.2.1

Divide each term in $3 (35 + 8 x) \geq 0$ by $3$ and simplify.

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Step 4.3.2.1.1

Divide each term in $3 (35 + 8 x) \geq 0$ by $3$ .

$\frac{3 (35 + 8 x)}{3} \geq \frac{0}{3}$

Step 4.3.2.1.2

Simplify the left side.

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Step 4.3.2.1.2.1

Cancel the common factor of $3$ .

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Step 4.3.2.1.2.1.1

Cancel the common factor.

$\frac{3 (35 + 8 x)}{3} \geq \frac{0}{3}$

Step 4.3.2.1.2.1.2

Divide $35 + 8 x$ by $1$ .

$35 + 8 x \geq \frac{0}{3}$

Step 4.3.2.1.3

Simplify the right side.

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Step 4.3.2.1.3.1

Divide $0$ by $3$ .

$35 + 8 x \geq 0$

Step 4.3.2.2

Subtract $35$ from both sides of the inequality.

$8 x \geq - 35$

Step 4.3.2.3

Divide each term in $8 x \geq - 35$ by $8$ and simplify.

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Step 4.3.2.3.1

Divide each term in $8 x \geq - 35$ by $8$ .

$\frac{8 x}{8} \geq \frac{- 35}{8}$

Step 4.3.2.3.2

Simplify the left side.

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Step 4.3.2.3.2.1

Cancel the common factor of $8$ .

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Step 4.3.2.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{- 35}{8}$

Step 4.3.2.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 35}{8}$

Step 4.3.2.3.3

Simplify the right side.

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Step 4.3.2.3.3.1

Move the negative in front of the fraction.

$x \geq - \frac{35}{8}$

Step 4.3.3

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

$[- \frac{35}{8}, \infty)$

Step 4.4

Find the domain of $f (x) = 6 x^{2} - 9 x - 1$ .

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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 $f^{- 1} (x) = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}, \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$ is the range of $f (x) = 6 x^{2} - 9 x - 1$ and the range of $f^{- 1} (x) = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}, \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$ is the domain of $f (x) = 6 x^{2} - 9 x - 1$ , then $f^{- 1} (x) = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}, \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$ is the inverse of $f (x) = 6 x^{2} - 9 x - 1$ .

$f^{- 1} (x) = \frac{9 + \sqrt{3 (35 + 8 x)}}{12}, \frac{9 - \sqrt{3 (35 + 8 x)}}{12}$

Step 5