Trigonometry Examples

$g (x) = - 4 x^{2} - 8$

Step 1

Write $g (x) = - 4 x^{2} - 8$ as an equation.

$y = - 4 x^{2} - 8$

Step 2

Interchange the variables.

$x = - 4 y^{2} - 8$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 4 y^{2} - 8 = x$ .

$- 4 y^{2} - 8 = x$

Step 3.2

Add $8$ to both sides of the equation.

$- 4 y^{2} = x + 8$

Step 3.3

Divide each term in $- 4 y^{2} = x + 8$ by $- 4$ and simplify.

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Step 3.3.1

Divide each term in $- 4 y^{2} = x + 8$ by $- 4$ .

$\frac{- 4 y^{2}}{- 4} = \frac{x}{- 4} + \frac{8}{- 4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 4$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 4 y^{2}}{- 4} = \frac{x}{- 4} + \frac{8}{- 4}$

Step 3.3.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{x}{- 4} + \frac{8}{- 4}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative in front of the fraction.

$y^{2} = - \frac{x}{4} + \frac{8}{- 4}$

Step 3.3.3.1.2

Divide $8$ by $- 4$ .

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

Factor $- 1$ out of $- \frac{x}{4} - 2$ .

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Step 3.5.1.1

Factor $- 1$ out of $- \frac{x}{4}$ .

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

Step 3.5.1.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 3.5.1.3

Factor $- 1$ out of $- (\frac{x}{4}) - 1 (2)$ .

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

Step 3.5.2

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

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

Step 3.5.3

Combine $2$ and $\frac{4}{4}$ .

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

Step 3.5.4

Combine the numerators over the common denominator.

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

Step 3.5.5

Multiply $2$ by $4$ .

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

Step 3.5.6

Rewrite $- \frac{x + 8}{4}$ as ${(\frac{1^{2}}{2})}^{2} \cdot (- (x + 8))$ .

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Step 3.5.6.1

Factor the perfect power $1^{2}$ out of $x + 8$ .

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

Step 3.5.6.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 3.5.6.3

Rearrange the fraction $\frac{1^{2} (x + 8)}{2^{2} \cdot 1}$ .

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

Step 3.5.6.4

Reorder $- 1$ and ${(\frac{1}{2})}^{2}$ .

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

Step 3.5.6.5

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

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

Step 3.5.6.6

Add parentheses.

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

Step 3.5.7

Pull terms out from under the radical.

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

Step 3.5.8

One to any power is one.

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

Step 3.5.9

Combine $\frac{1}{2}$ and $\sqrt{- (x + 8)}$ .

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

Step 3.6

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

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Step 3.6.1

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

$y = \frac{\sqrt{- (x + 8)}}{2}$

Step 3.6.2

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

$y = - \frac{\sqrt{- (x + 8)}}{2}$

Step 3.6.3

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

$y = \frac{\sqrt{- (x + 8)}}{2}$

$y = - \frac{\sqrt{- (x + 8)}}{2}$

$y = \frac{\sqrt{- (x + 8)}}{2}$

$y = - \frac{\sqrt{- (x + 8)}}{2}$

$y = \frac{\sqrt{- (x + 8)}}{2}$

$y = - \frac{\sqrt{- (x + 8)}}{2}$

Step 4

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

$g^{- 1} (x) = \frac{\sqrt{- (x + 8)}}{2}, - \frac{\sqrt{- (x + 8)}}{2}$

Step 5

Verify if $g^{- 1} (x) = \frac{\sqrt{- (x + 8)}}{2}, - \frac{\sqrt{- (x + 8)}}{2}$ is the inverse of $g (x) = - 4 x^{2} - 8$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $g (x) = - 4 x^{2} - 8$ and $g^{- 1} (x) = \frac{\sqrt{- (x + 8)}}{2}, - \frac{\sqrt{- (x + 8)}}{2}$ and compare them.

Step 5.2

Find the range of $g (x) = - 4 x^{2} - 8$ .

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Step 5.2.1

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

Interval Notation:

$(- \infty, - 8]$

Step 5.3

Find the domain of $\frac{\sqrt{- (x + 8)}}{2}$ .

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Step 5.3.1

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

$- (x + 8) \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Simplify $- (x + 8)$ .

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Step 5.3.2.1.1

Apply the distributive property.

$- x - 1 \cdot 8 \geq 0$

Step 5.3.2.1.2

Multiply $- 1$ by $8$ .

$- x - 8 \geq 0$

Step 5.3.2.2

Add $8$ to both sides of the inequality.

$- x \geq 8$

Step 5.3.2.3

Divide each term in $- x \geq 8$ by $- 1$ and simplify.

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Step 5.3.2.3.1

Divide each term in $- x \geq 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{8}{- 1}$

Step 5.3.2.3.2

Simplify the left side.

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Step 5.3.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{8}{- 1}$

Step 5.3.2.3.2.2

Divide $x$ by $1$ .

$x \leq \frac{8}{- 1}$

Step 5.3.2.3.3

Simplify the right side.

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Step 5.3.2.3.3.1

Divide $8$ by $- 1$ .

$x \leq - 8$

Step 5.3.3

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

$(- \infty, - 8]$

Step 5.4

Find the domain of $g (x) = - 4 x^{2} - 8$ .

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Step 5.4.1

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 5.5

Since the domain of $g^{- 1} (x) = \frac{\sqrt{- (x + 8)}}{2}, - \frac{\sqrt{- (x + 8)}}{2}$ is the range of $g (x) = - 4 x^{2} - 8$ and the range of $g^{- 1} (x) = \frac{\sqrt{- (x + 8)}}{2}, - \frac{\sqrt{- (x + 8)}}{2}$ is the domain of $g (x) = - 4 x^{2} - 8$ , then $g^{- 1} (x) = \frac{\sqrt{- (x + 8)}}{2}, - \frac{\sqrt{- (x + 8)}}{2}$ is the inverse of $g (x) = - 4 x^{2} - 8$ .

$g^{- 1} (x) = \frac{\sqrt{- (x + 8)}}{2}, - \frac{\sqrt{- (x + 8)}}{2}$

Step 6