Find the Inverse 6x^4

$6 x^{4}$

Step 1

Interchange the variables.

$x = 6 y^{4}$

Step 2

Solve for $y$ .

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

$6 y^{4} = x$

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

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

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

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

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

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

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

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

Rewrite $\sqrt[4]{\frac{x}{6}}$ as $\frac{\sqrt[4]{x}}{\sqrt[4]{6}}$ .

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

Multiply $\frac{\sqrt[4]{x}}{\sqrt[4]{6}}$ by $\frac{{\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{x}}{\sqrt[4]{6}} \cdot \frac{{\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{3}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt[4]{x}}{\sqrt[4]{6}}$ by $\frac{{\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{3}}$ .

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{\sqrt[4]{6} {\sqrt[4]{6}}^{3}}$

Raise $\sqrt[4]{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{1} {\sqrt[4]{6}}^{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{1 + 3}}$

Add $1$ and $3$ .

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{{\sqrt[4]{6}}^{4}}$

Rewrite ${\sqrt[4]{6}}^{4}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{6}$ as $6^{\frac{1}{4}}$ .

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{{(6^{\frac{1}{4}})}^{4}}$

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

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{6^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{6^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{6^{\frac{4}{4}}}$

Rewrite the expression.

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{6^{1}}$

Evaluate the exponent.

$y = \pm \frac{\sqrt[4]{x} {\sqrt[4]{6}}^{3}}{6}$

Simplify the numerator.

Rewrite ${\sqrt[4]{6}}^{3}$ as $\sqrt[4]{6^{3}}$ .

$y = \pm \frac{\sqrt[4]{x} \sqrt[4]{6^{3}}}{6}$

Raise $6$ to the power of $3$ .

$y = \pm \frac{\sqrt[4]{x} \sqrt[4]{216}}{6}$

Simplify with factoring out.

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt[4]{x \cdot 216}}{6}$

Reorder factors in $\pm \frac{\sqrt[4]{x \cdot 216}}{6}$ .

$y = \pm \frac{\sqrt[4]{216 x}}{6}$

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.

$y = \frac{\sqrt[4]{216 x}}{6}$

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

$y = - \frac{\sqrt[4]{216 x}}{6}$

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

$y = \frac{\sqrt[4]{216 x}}{6}$

$y = - \frac{\sqrt[4]{216 x}}{6}$

$y = \frac{\sqrt[4]{216 x}}{6}$

$y = - \frac{\sqrt[4]{216 x}}{6}$

$y = \frac{\sqrt[4]{216 x}}{6}$

$y = - \frac{\sqrt[4]{216 x}}{6}$

Step 3

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

$f^{- 1} (x) = \frac{\sqrt[4]{216 x}}{6}, - \frac{\sqrt[4]{216 x}}{6}$

Step 4

Verify if $f^{- 1} (x) = \frac{\sqrt[4]{216 x}}{6}, - \frac{\sqrt[4]{216 x}}{6}$ is the inverse of $f (x) = 6 x^{4}$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 6 x^{4}$ and $f^{- 1} (x) = \frac{\sqrt[4]{216 x}}{6}, - \frac{\sqrt[4]{216 x}}{6}$ and compare them.

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

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

Interval Notation:

$[0, \infty)$

Find the domain of $\frac{\sqrt[4]{216 x}}{6}$ .

Set the radicand in $\sqrt[4]{216 x}$ greater than or equal to $0$ to find where the expression is defined.

$216 x \geq 0$

Divide each term in $216 x \geq 0$ by $216$ and simplify.

Divide each term in $216 x \geq 0$ by $216$ .

$\frac{216 x}{216} \geq \frac{0}{216}$

Simplify the left side.

Cancel the common factor of $216$ .

Cancel the common factor.

$\frac{216 x}{216} \geq \frac{0}{216}$

Divide $x$ by $1$ .

$x \geq \frac{0}{216}$

Simplify the right side.

Divide $0$ by $216$ .

$x \geq 0$

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

$[0, \infty)$

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

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 $f^{- 1} (x) = \frac{\sqrt[4]{216 x}}{6}, - \frac{\sqrt[4]{216 x}}{6}$ is the range of $f (x) = 6 x^{4}$ and the range of $f^{- 1} (x) = \frac{\sqrt[4]{216 x}}{6}, - \frac{\sqrt[4]{216 x}}{6}$ is the domain of $f (x) = 6 x^{4}$ , then $f^{- 1} (x) = \frac{\sqrt[4]{216 x}}{6}, - \frac{\sqrt[4]{216 x}}{6}$ is the inverse of $f (x) = 6 x^{4}$ .

$f^{- 1} (x) = \frac{\sqrt[4]{216 x}}{6}, - \frac{\sqrt[4]{216 x}}{6}$

Step 5

Find the Inverse 6x^4

Related Questions

Support LCMGCF.com by disabling your adblocker.