Trigonometry Examples

${(y - 2)}^{2} = 3 (x + 1)$

Step 1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y - 2 = \pm \sqrt{3 (x + 1)}$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

$y - 2 = \sqrt{3 (x + 1)}$

Step 2.2

Add $2$ to both sides of the equation.

$y = \sqrt{3 (x + 1)} + 2$

Step 2.3

Next, use the negative value of the $\pm$ to find the second solution.

$y - 2 = - \sqrt{3 (x + 1)}$

Step 2.4

Add $2$ to both sides of the equation.

$y = - \sqrt{3 (x + 1)} + 2$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{3 (x + 1)} + 2$

$y = - \sqrt{3 (x + 1)} + 2$

$y = \sqrt{3 (x + 1)} + 2$

$y = - \sqrt{3 (x + 1)} + 2$

Step 3

Interchange the variables. Create an equation for each expression.

$x = \sqrt{3 (y + 1)} + 2, x = - \sqrt{3 (y + 1)} + 2$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $\sqrt{3 (y + 1)} + 2 = x$ .

$\sqrt{3 (y + 1)} + 2 = x$

Step 4.2

Subtract $2$ from both sides of the equation.

$\sqrt{3 (y + 1)} = x - 2$

Step 4.3

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

${\sqrt{3 (y + 1)}}^{2} = {(x - 2)}^{2}$

Step 4.4

Simplify each side of the equation.

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Step 4.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (y + 1)}$ as ${(3 (y + 1))}^{\frac{1}{2}}$ .

${({(3 (y + 1))}^{\frac{1}{2}})}^{2} = {(x - 2)}^{2}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Simplify ${({(3 (y + 1))}^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1.1

Multiply the exponents in ${({(3 (y + 1))}^{\frac{1}{2}})}^{2}$ .

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Step 4.4.2.1.1.1

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

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

Step 4.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.4.2.1.1.2.1

Cancel the common factor.

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

Step 4.4.2.1.1.2.2

Rewrite the expression.

${(3 (y + 1))}^{1} = {(x - 2)}^{2}$

Step 4.4.2.1.2

Apply the distributive property.

${(3 y + 3 \cdot 1)}^{1} = {(x - 2)}^{2}$

Step 4.4.2.1.3

Simplify.

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Step 4.4.2.1.3.1

Multiply $3$ by $1$ .

${(3 y + 3)}^{1} = {(x - 2)}^{2}$

Step 4.4.2.1.3.2

Simplify.

$3 y + 3 = {(x - 2)}^{2}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

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

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Step 4.4.3.1.1

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

$3 y + 3 = (x - 2) (x - 2)$

Step 4.4.3.1.2

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

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Step 4.4.3.1.2.1

Apply the distributive property.

$3 y + 3 = x (x - 2) - 2 (x - 2)$

Step 4.4.3.1.2.2

Apply the distributive property.

$3 y + 3 = x \cdot x + x \cdot - 2 - 2 (x - 2)$

Step 4.4.3.1.2.3

Apply the distributive property.

$3 y + 3 = x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 4.4.3.1.3

Simplify and combine like terms.

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Step 4.4.3.1.3.1

Simplify each term.

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Step 4.4.3.1.3.1.1

Multiply $x$ by $x$ .

$3 y + 3 = x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2$

Step 4.4.3.1.3.1.2

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

$3 y + 3 = x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2$

Step 4.4.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$3 y + 3 = x^{2} - 2 x - 2 x + 4$

Step 4.4.3.1.3.2

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

$3 y + 3 = x^{2} - 4 x + 4$

Step 4.5

Solve for $y$ .

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Step 4.5.1

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

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Step 4.5.1.1

Subtract $3$ from both sides of the equation.

$3 y = x^{2} - 4 x + 4 - 3$

Step 4.5.1.2

Subtract $3$ from $4$ .

$3 y = x^{2} - 4 x + 1$

Step 4.5.2

Divide each term in $3 y = x^{2} - 4 x + 1$ by $3$ and simplify.

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Step 4.5.2.1

Divide each term in $3 y = x^{2} - 4 x + 1$ by $3$ .

$\frac{3 y}{3} = \frac{x^{2}}{3} + \frac{- 4 x}{3} + \frac{1}{3}$

Step 4.5.2.2

Simplify the left side.

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Step 4.5.2.2.1

Cancel the common factor of $3$ .

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Step 4.5.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{x^{2}}{3} + \frac{- 4 x}{3} + \frac{1}{3}$

Step 4.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{3} + \frac{- 4 x}{3} + \frac{1}{3}$

Step 4.5.2.3

Simplify the right side.

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Step 4.5.2.3.1

Move the negative in front of the fraction.

$y = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$

Step 5

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

$f^{- 1} (x) = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$

Step 6

Verify if $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$ is the inverse of $f (x) = \sqrt{3 (x + 1)} + 2, - \sqrt{3 (x + 1)} + 2$ .

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Step 6.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \sqrt{3 (x + 1)} + 2, - \sqrt{3 (x + 1)} + 2$ and $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$ and compare them.

Step 6.2

Find the range of $f (x) = \sqrt{3 (x + 1)} + 2, - \sqrt{3 (x + 1)} + 2$ .

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Step 6.2.1

Find the range of $\sqrt{3 (x + 1)} + 2$ .

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Step 6.2.1.1

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

Interval Notation:

$[2, \infty)$

Step 6.2.2

Find the range of $- \sqrt{3 (x + 1)} + 2$ .

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Step 6.2.2.1

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

Interval Notation:

$(- \infty, 2]$

Step 6.2.3

Find the union of $[2, \infty), (- \infty, 2]$ .

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Step 6.2.3.1

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

$(- \infty, \infty)$

Step 6.3

Find the domain of $\sqrt{3 (x + 1)} + 2$ .

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Step 6.3.1

Set the radicand in $\sqrt{3 (x + 1)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (x + 1) \geq 0$

Step 6.3.2

Solve for $x$ .

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Step 6.3.2.1

Divide each term in $3 (x + 1) \geq 0$ by $3$ and simplify.

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Step 6.3.2.1.1

Divide each term in $3 (x + 1) \geq 0$ by $3$ .

$\frac{3 (x + 1)}{3} \geq \frac{0}{3}$

Step 6.3.2.1.2

Simplify the left side.

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Step 6.3.2.1.2.1

Cancel the common factor of $3$ .

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Step 6.3.2.1.2.1.1

Cancel the common factor.

$\frac{3 (x + 1)}{3} \geq \frac{0}{3}$

Step 6.3.2.1.2.1.2

Divide $x + 1$ by $1$ .

$x + 1 \geq \frac{0}{3}$

Step 6.3.2.1.3

Simplify the right side.

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Step 6.3.2.1.3.1

Divide $0$ by $3$ .

$x + 1 \geq 0$

Step 6.3.2.2

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

Step 6.3.3

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

$[- 1, \infty)$

Step 6.4

Since the domain of $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$ is the range of $f (x) = \sqrt{3 (x + 1)} + 2, - \sqrt{3 (x + 1)} + 2$ and the range of $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$ is the domain of $f (x) = \sqrt{3 (x + 1)} + 2, - \sqrt{3 (x + 1)} + 2$ , then $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$ is the inverse of $f (x) = \sqrt{3 (x + 1)} + 2, - \sqrt{3 (x + 1)} + 2$ .

$f^{- 1} (x) = \frac{x^{2}}{3} - \frac{4 x}{3} + \frac{1}{3}$

Step 7