Trigonometry Examples

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

Step 1

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

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

Step 2

Subtract $x$ from both sides of the equation.

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 2$ , and $c = - x$ into the quadratic formula and solve for $y$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 5.1.3

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

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Step 5.1.3.1

Factor $4$ out of $4$ .

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

Step 5.1.3.2

Factor $4$ out of $4 \cdot 1 + 4 x$ .

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

Step 5.1.4

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

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Step 5.1.4.1

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

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

Step 5.1.4.2

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

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

Step 5.1.5

Pull terms out from under the radical.

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

Step 5.1.6

One to any power is one.

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

Step 5.2

Multiply $2$ by $1$ .

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

Step 5.3

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

$y = 1 \pm \sqrt{1 + x}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 6.1.3

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

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Step 6.1.3.1

Factor $4$ out of $4$ .

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

Step 6.1.3.2

Factor $4$ out of $4 \cdot 1 + 4 x$ .

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

Step 6.1.4

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

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Step 6.1.4.1

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

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

Step 6.1.4.2

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

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

Step 6.1.5

Pull terms out from under the radical.

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

Step 6.1.6

One to any power is one.

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

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

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

$y = 1 \pm \sqrt{1 + x}$

Step 6.4

Change the $\pm$ to $+$ .

$y = 1 + \sqrt{1 + x}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 7.1.2.2

Multiply $- 4$ by $- 1$ .

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

Step 7.1.3

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

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Step 7.1.3.1

Factor $4$ out of $4$ .

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

Step 7.1.3.2

Factor $4$ out of $4 \cdot 1 + 4 x$ .

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

Step 7.1.4

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

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Step 7.1.4.1

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

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

Step 7.1.4.2

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

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

Step 7.1.5

Pull terms out from under the radical.

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

Step 7.1.6

One to any power is one.

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

Step 7.2

Multiply $2$ by $1$ .

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

Step 7.3

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

$y = 1 \pm \sqrt{1 + x}$

Step 7.4

Change the $\pm$ to $-$ .

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

Step 8

The final answer is the combination of both solutions.

$y = 1 + \sqrt{1 + x}$

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

Step 9

Interchange the variables. Create an equation for each expression.

$x = 1 + \sqrt{1 + y}, x = 1 - \sqrt{1 + y}$

Step 10

Solve for $y$ .

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Step 10.1

Rewrite the equation as $1 + \sqrt{1 + y} = x$ .

$1 + \sqrt{1 + y} = x$

Step 10.2

Subtract $1$ from both sides of the equation.

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

Step 10.3

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

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

Step 10.4

Simplify each side of the equation.

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Step 10.4.1

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

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

Step 10.4.2

Simplify the left side.

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Step 10.4.2.1

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

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Step 10.4.2.1.1

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

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Step 10.4.2.1.1.1

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

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

Step 10.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 10.4.2.1.1.2.1

Cancel the common factor.

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

Step 10.4.2.1.1.2.2

Rewrite the expression.

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

Step 10.4.2.1.2

Simplify.

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

Step 10.4.3

Simplify the right side.

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Step 10.4.3.1

Simplify ${(x - 1)}^{2}$ .

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Step 10.4.3.1.1

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

$1 + y = (x - 1) (x - 1)$

Step 10.4.3.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 10.4.3.1.2.1

Apply the distributive property.

$1 + y = x (x - 1) - 1 (x - 1)$

Step 10.4.3.1.2.2

Apply the distributive property.

$1 + y = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Step 10.4.3.1.2.3

Apply the distributive property.

$1 + y = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 10.4.3.1.3

Simplify and combine like terms.

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Step 10.4.3.1.3.1

Simplify each term.

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Step 10.4.3.1.3.1.1

Multiply $x$ by $x$ .

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

Step 10.4.3.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 10.4.3.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 10.4.3.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 10.4.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 10.4.3.1.3.2

Subtract $x$ from $- x$ .

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

Step 10.5

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

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Step 10.5.1

Subtract $1$ from both sides of the equation.

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

Step 10.5.2

Combine the opposite terms in $x^{2} - 2 x + 1 - 1$ .

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Step 10.5.2.1

Subtract $1$ from $1$ .

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

Step 10.5.2.2

Add $x^{2} - 2 x$ and $0$ .

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

Step 11

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

$f^{- 1} (x) = x^{2} - 2 x$

Step 12

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

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Step 12.1

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

Step 12.2

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

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Step 12.2.1

Find the range of $1 + \sqrt{1 + x}$ .

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Step 12.2.1.1

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

Interval Notation:

$[1, \infty)$

Step 12.2.2

Find the range of $1 - \sqrt{1 + x}$ .

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Step 12.2.2.1

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

Interval Notation:

$(- \infty, 1]$

Step 12.2.3

Find the union of $[1, \infty), (- \infty, 1]$ .

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Step 12.2.3.1

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

$(- \infty, \infty)$

Step 12.3

Find the domain of $1 + \sqrt{1 + x}$ .

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Step 12.3.1

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

$1 + x \geq 0$

Step 12.3.2

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

Step 12.3.3

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

$[- 1, \infty)$

Step 12.4

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

$f^{- 1} (x) = x^{2} - 2 x$

Step 13