Find the Inverse y=-4x^2-2

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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

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

Add $2$ to both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Cancel the common factor of $2$ and $- 4$ .

Factor $2$ out of $2$ .

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

Cancel the common factors.

Factor $2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

Simplify $\pm \sqrt{- \frac{x}{4} - \frac{1}{2}}$ .

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

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

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

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

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

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

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

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

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

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

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

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

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

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

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

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

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

One to any power is one.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Step 3

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

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

Step 4

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

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

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

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

Interval Notation:

$(- \infty, - 2]$

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

Set the radicand in $\sqrt{- (x + 2)}$ greater than or equal to $0$ to find where the expression is defined.

$- (x + 2) \geq 0$

Solve for $x$ .

Simplify $- (x + 2)$ .

Apply the distributive property.

$- x - 1 \cdot 2 \geq 0$

Multiply $- 1$ by $2$ .

$- x - 2 \geq 0$

Add $2$ to both sides of the inequality.

$- x \geq 2$

Divide each term in $- x \geq 2$ by $- 1$ and simplify.

Divide each term in $- x \geq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{2}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{2}{- 1}$

Simplify the right side.

Divide $2$ by $- 1$ .

$x \leq - 2$

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

$(- \infty, - 2]$

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

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.

$(- \infty, \infty)$

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

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

Step 5

Find the Inverse y=-4x^2-2

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