Find the Inverse f(x)=-6x^4-1

$f (x) = - 6 x^{4} - 1$

Step 1

Write $f (x) = - 6 x^{4} - 1$ as an equation.

$y = - 6 x^{4} - 1$

Step 2

Interchange the variables.

$x = - 6 y^{4} - 1$

Step 3

Solve for $y$ .

Rewrite the equation as $- 6 y^{4} - 1 = x$ .

$- 6 y^{4} - 1 = x$

Add $1$ to both sides of the equation.

$- 6 y^{4} = x + 1$

Divide each term in $- 6 y^{4} = x + 1$ by $- 6$ and simplify.

Divide each term in $- 6 y^{4} = x + 1$ by $- 6$ .

$\frac{- 6 y^{4}}{- 6} = \frac{x}{- 6} + \frac{1}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 y^{4}}{- 6} = \frac{x}{- 6} + \frac{1}{- 6}$

Divide $y^{4}$ by $1$ .

$y^{4} = \frac{x}{- 6} + \frac{1}{- 6}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y^{4} = - \frac{x}{6} + \frac{1}{- 6}$

Move the negative in front of the fraction.

$y^{4} = - \frac{x}{6} - \frac{1}{6}$

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

$y = \pm \sqrt[4]{- \frac{x}{6} - \frac{1}{6}}$

Simplify $\pm \sqrt[4]{- \frac{x}{6} - \frac{1}{6}}$ .

Factor $- 1$ out of $- \frac{x}{6} - \frac{1}{6}$ .

Factor $- 1$ out of $- \frac{x}{6}$ .

$y = \pm \sqrt[4]{- (\frac{x}{6}) - \frac{1}{6}}$

Factor $- 1$ out of $- \frac{1}{6}$ .

$y = \pm \sqrt[4]{- (\frac{x}{6}) - (\frac{1}{6})}$

Factor $- 1$ out of $- (\frac{x}{6}) - (\frac{1}{6})$ .

$y = \pm \sqrt[4]{- (\frac{x}{6} + \frac{1}{6})}$

Combine the numerators over the common denominator.

$y = \pm \sqrt[4]{- \frac{x + 1}{6}}$

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.

$y = \sqrt[4]{- \frac{x + 1}{6}}$

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

$y = - \sqrt[4]{- \frac{x + 1}{6}}$

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

$y = \sqrt[4]{- \frac{x + 1}{6}}$

$y = - \sqrt[4]{- \frac{x + 1}{6}}$

$y = \sqrt[4]{- \frac{x + 1}{6}}$

$y = - \sqrt[4]{- \frac{x + 1}{6}}$

$y = \sqrt[4]{- \frac{x + 1}{6}}$

$y = - \sqrt[4]{- \frac{x + 1}{6}}$

Step 4

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

$f^{- 1} (x) = \sqrt[4]{- \frac{x + 1}{6}}, - \sqrt[4]{- \frac{x + 1}{6}}$

Step 5

Verify if $f^{- 1} (x) = \sqrt[4]{- \frac{x + 1}{6}}, - \sqrt[4]{- \frac{x + 1}{6}}$ is the inverse of $f (x) = - 6 x^{4} - 1$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = - 6 x^{4} - 1$ and $f^{- 1} (x) = \sqrt[4]{- \frac{x + 1}{6}}, - \sqrt[4]{- \frac{x + 1}{6}}$ and compare them.

Find the range of $f (x) = - 6 x^{4} - 1$ .

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

Interval Notation:

$(- \infty, - 1]$

Find the domain of $\sqrt[4]{- \frac{x + 1}{6}}$ .

Set the radicand in $\sqrt[4]{- \frac{x + 1}{6}}$ greater than or equal to $0$ to find where the expression is defined.

$- \frac{x + 1}{6} \geq 0$

Solve for $x$ .

Divide each term in $- \frac{x + 1}{6} \geq 0$ by $- 1$ and simplify.

Divide each term in $- \frac{x + 1}{6} \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x + 1}{6}}{- 1} \leq \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{x + 1}{6}}{1} \leq \frac{0}{- 1}$

Divide $\frac{x + 1}{6}$ by $1$ .

$\frac{x + 1}{6} \leq \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$\frac{x + 1}{6} \leq 0$

Multiply both sides by $6$ .

$\frac{x + 1}{6} \cdot 6 \leq 0 \cdot 6$

Simplify.

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{x + 1}{6} \cdot 6 \leq 0 \cdot 6$

Rewrite the expression.

$x + 1 \leq 0 \cdot 6$

Simplify the right side.

Multiply $0$ by $6$ .

$x + 1 \leq 0$

Subtract $1$ from both sides of the inequality.

$x \leq - 1$

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

$(- \infty, - 1]$

Find the domain of $f (x) = - 6 x^{4} - 1$ .

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.

$(- \infty, \infty)$

Since the domain of $f^{- 1} (x) = \sqrt[4]{- \frac{x + 1}{6}}, - \sqrt[4]{- \frac{x + 1}{6}}$ is the range of $f (x) = - 6 x^{4} - 1$ and the range of $f^{- 1} (x) = \sqrt[4]{- \frac{x + 1}{6}}, - \sqrt[4]{- \frac{x + 1}{6}}$ is the domain of $f (x) = - 6 x^{4} - 1$ , then $f^{- 1} (x) = \sqrt[4]{- \frac{x + 1}{6}}, - \sqrt[4]{- \frac{x + 1}{6}}$ is the inverse of $f (x) = - 6 x^{4} - 1$ .

$f^{- 1} (x) = \sqrt[4]{- \frac{x + 1}{6}}, - \sqrt[4]{- \frac{x + 1}{6}}$

Step 6

Find the Inverse f(x)=-6x^4-1

Related Questions

Support LCMGCF.com by disabling your adblocker.