Find the Inverse f(x)=x^4+7 for x>=0

$f (x) = x^{4} + 7$ for $x \geq 0$

Step 1

Find the range of the given function.

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

$[7, \infty)$

Convert $[7, \infty)$ to an inequality.

$y \geq 7$

Step 2

Find the inverse.

Interchange the variables.

$x = y^{4} + 7$

Solve for $y$ .

Rewrite the equation as $y^{4} + 7 = x$ .

$y^{4} + 7 = x$

Subtract $7$ from both sides of the equation.

$y^{4} = x - 7$

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt[4]{x - 7}$

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.

$y = \sqrt[4]{x - 7}$

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

$y = - \sqrt[4]{x - 7}$

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

$y = \sqrt[4]{x - 7}$

$y = - \sqrt[4]{x - 7}$

$y = \sqrt[4]{x - 7}$

$y = - \sqrt[4]{x - 7}$

$y = \sqrt[4]{x - 7}$

$y = - \sqrt[4]{x - 7}$

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

$f^{- 1} (x) = \sqrt[4]{x - 7}, - \sqrt[4]{x - 7}$

Step 3

Find the inverse using the domain and the range of the original function.

$f^{- 1} (x) = \sqrt[4]{x - 7}, x \geq 7$

Step 4